Function Definitions, Evaluation, and Custom Operators Alıştırma Soruları — GMAT

Soru 1Soru →

For all real numbers $x$ and $y$ , the custom operation $\odot$ is defined by $x \odot y = x^2 - 2y$ . What is the value of $(3 \odot 2) \odot 4$ ?

17

6

-4

9

-7

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

Following the explicit operator rule

x \odot y = x^2 - 2y

, first calculate the inner expression

(3 \odot 2) = 3^2 - 2(2) = 5

. Next, substitute 5 into the outer expression to compute

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

. Thus, the value 17 is correct.

Adım Adım Çözüm

1

Evaluate the inner function expression

(3 \odot 2)

using the rule

x \odot y = x^2 - 2y

3 \odot 2 = 3^2 - 2(2) = 9 - 4 = 5

Parentheses dictate that inner custom operations must be evaluated first.

2

Substitute the result of step 1 back into the expression to evaluate

5 \odot 4

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

Applying the custom operator rule with

x = 5

and

y = 4

computes the final target value.

Anahtar Kavram

Custom Operator Evaluation and Order of Operations

Tahmini Süre:1m 15s

Soru 2Soru →

For all real numbers

x

except

0

and

1

, the function

f

satisfies the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x + 3

What is the value of

f(2)

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

f(2)

is

7

.

Evaluating the functional equation iteratively at the points of the cycle

x = 2

,

x = -1

, and

x = \frac{1}{2}

produces a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this linear system gives

f(2) = 7

.

Adım Adım Çözüm

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) + 3 \implies f(2) + 2f(-1) = 21

This establishes a relationship between

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) + 3 \implies f(-1) + 2f\left(\frac{1}{2}\right) = -6

This links

f(-1)

to

f\left(\frac{1}{2}\right)

to build a closed system of transformations under the mapping

g(x) = \frac{1}{1-x}

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) + 3 \implies f\left(\frac{1}{2}\right) + 2f(2) = \frac{15}{2}

This completes the cycle since

g\left(\frac{1}{2}\right) = 2

, creating a system of three linear equations with three unknown values.

4

Solve the system of three linear equations for

f(2)

Let

a = f(2)

,

b = f(-1)

, and

c = f\left(\frac{1}{2}\right)

. From equation (3),

c = \frac{15}{2} - 2a

. Substituting

c

into equation (2) gives

b + 2\left(\frac{15}{2} - 2a\right) = -6 \implies b = 4a - 21

. Substituting

b

into equation (1) gives

a + 2(4a - 21) = 21 \implies 9a - 42 = 21 \implies 9a = 63 \implies a = 7

.

Algebraic substitution yields the exact value of

f(2)

.

Anahtar Kavram

Cyclic Functional Equations and System Substitution

Tahmini Süre:2m 30s

Soru 3Soru →

For all real numbers $a$ and $b$ , the custom operation $\star$ is defined by $a \star b = (a + b)^2 - 3b$ . What is the value of $4 \star 2$ ?

30

24

14

33

18

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

Substituting

a = 4

and

b = 2

into the definition

(a + b)^2 - 3b

gives

(4 + 2)^2 - 3(2) = 6^2 - 6 = 36 - 6 = 30

.

Adım Adım Çözüm

1

Identify the values assigned to the custom operator variables

a = 4

and

b = 2

The expression

4 \star 2

corresponds to evaluating

a \star b

where

a

is the first operand and

b

is the second operand.

2

Substitute

a = 4

and

b = 2

into the operator definition

(a + b)^2 - 3b

(4 + 2)^2 - 3(2)

Direct substitution into the given algebraic definition.

3

Perform the operations in parentheses and exponentiation

6^2 - 6 = 36 - 6

First add inside the parentheses (

4 + 2 = 6

), then square the sum (

6^2 = 36

), and multiply

3 \times 2 = 6

.

4

Subtract to find the final value

30

Subtracting

6

from

36

yields

30

.

Anahtar Kavram

Custom Operator Evaluation

Tahmini Süre:45s

Soru 4Soru →

For all non-zero real numbers $a$ and $b$ , the custom operator $\Delta$ is defined by $a \Delta b = \frac{a^2 - b^2}{ab}$ . A function $f(x)$ is defined for all $x \neq 0$ and $x \neq -1$ by $f(x) = x \Delta (x + 1)$ . If $k$ is a positive constant such that $f(k) = -\frac{7}{12}$ , what is the value of $(k + 2) \Delta k$ ?

-\frac{16}{15}

\frac{4}{15}

\frac{5}{6}

\frac{16}{15}

\frac{3}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{16}{15}$

Cevap

The value of

(k + 2) \Delta k

is

\frac{16}{15}

.

Applying the custom operator to

f(x)

yields

f(x) = \frac{x^2 - (x+1)^2}{x(x+1)} = \frac{-2x - 1}{x^2 + x}

. Setting

f(k) = -\frac{7}{12}

gives

\frac{2k + 1}{k^2 + k} = \frac{7}{12}

, which simplifies to

7k^2 - 17k - 12 = 0

. Factoring

(7k + 4)(k - 3) = 0

and selecting the positive root gives

k = 3

. Evaluating

(3+2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \cdot 3} = \frac{16}{15}

.

Adım Adım Çözüm

1

Express

f(x)

using the definition of the custom operator

\Delta

.

f(x) = x \Delta (x + 1) = \frac{x^2 - (x + 1)^2}{x(x + 1)} = \frac{x^2 - (x^2 + 2x + 1)}{x(x + 1)} = \frac{-2x - 1}{x^2 + x}

Substitute

a = x

and

b = x + 1

into

a \Delta b = \frac{a^2 - b^2}{ab}

.

2

Set

f(k) = -\frac{7}{12}

and solve for the positive constant

k

.

\frac{-2k - 1}{k^2 + k} = -\frac{7}{12} \implies \frac{2k + 1}{k^2 + k} = \frac{7}{12} \implies 12(2k + 1) = 7(k^2 + k) \implies 7k^2 - 17k - 12 = 0

Cross-multiply and rearrange into a standard quadratic equation.

3

Factor the quadratic equation to find the positive root

k

.

(7k + 4)(k - 3) = 0 \implies k = -\frac{4}{7}

or

k = 3

. Since

k > 0

,

k = 3

.

The question specifies that

k

is a positive constant.

4

Evaluate

(k + 2) \Delta k

for

k = 3

.

(3 + 2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \times 3} = \frac{25 - 9}{15} = \frac{16}{15}

Substitute

a = 5

and

b = 3

into the custom operator definition.

Anahtar Kavram

Function definition substitution, non-commutative custom operators, and quadratic solving.

Tahmini Süre:2m 30s

Soru 5Soru →

For all real numbers $x$ and $y$ , the custom operator $\Delta$ is defined by $x \Delta y = x^2 - 2y$ . If $k$ is a real number such that $(3 \Delta k) \Delta 4 = 17$ , what is the sum of all possible values of $k$ ?

1

2

8

9

14

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The sum of all possible values of

k

is 9.

Applying the custom operation rule

x \Delta y = x^2 - 2y

sequentially, the inner evaluation gives

3 \Delta k = 9 - 2k

. Applying the operation again to

(9 - 2k) \Delta 4

gives

(9 - 2k)^2 - 8 = 17

, which simplifies to

(9 - 2k)^2 = 25

. Taking the square root gives two valid solutions for

9 - 2k

:

5

and

-5

. Solving

9 - 2k = 5

yields

k = 2

, and solving

9 - 2k = -5

yields

k = 7

. The sum of these two values is

2 + 7 = 9

.

Adım Adım Çözüm

1

Evaluate the inner custom operation

3 \Delta k

.

3 \Delta k = 3^2 - 2k = 9 - 2k

Apply the definition

x \Delta y = x^2 - 2y

with

x = 3

and

y = k

.

2

Substitute the inner result into the outer expression

(9 - 2k) \Delta 4 = 17

.

(9 - 2k)^2 - 2(4) = 17 \implies (9 - 2k)^2 - 8 = 17

Apply the definition

x \Delta y = x^2 - 2y

with

x = 9 - 2k

and

y = 4

.

3

Isolate the squared term and solve for

9 - 2k

.

(9 - 2k)^2 = 25 \implies 9 - 2k = 5

or

9 - 2k = -5

Adding 8 to both sides gives 25; taking the square root requires considering both positive and negative roots.

4

Solve each linear equation for

k

and calculate their sum.

Case 1:

9 - 2k = 5 \implies 2k = 4 \implies k = 2

.
Case 2:

9 - 2k = -5 \implies 2k = 14 \implies k = 7

.
Sum =

2 + 7 = 9

.

Solving both equations yields all possible values for

k

.

Anahtar Kavram

Custom Operator Evaluation and Quadratic Equation Root Extraction

Tahmini Süre:2m 0s

Soru 6Soru →

For all real numbers $p$ and $q$ , the operation $\nabla$ is defined by $p \nabla q = 2p + 3q - pq$ . What is the value of $3 \nabla 4$ ?

5

6

7

18

30

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

3 \nabla 4

is 6.

Substituting

p = 3

and

q = 4

directly into the expression

2p + 3q - pq

gives

2(3) + 3(4) - (3)(4) = 6 + 12 - 12 = 6

.

Adım Adım Çözüm

1

Identify the assigned values for the operator inputs.

p = 3

and

q = 4

.

The expression

3 \nabla 4

matches

p \nabla q

with

p

as the first argument and

q

as the second.

2

Substitute

p = 3

and

q = 4

into the custom operation formula

2p + 3q - pq

.

2(3) + 3(4) - (3)(4)

Direct evaluation of the defined function rule.

3

Perform the multiplication and algebraic addition/subtraction.

6 + 12 - 12 = 6

Simplifying numerical operations gives the final result.

Anahtar Kavram

Custom Operator Evaluation

Soru 7Soru →

For all real numbers $x$ and $y$ , the custom operation $\diamondsuit$ is defined by $x \diamondsuit y = 3x - 2y + 1$ . What is the value of $5 \diamondsuit 4$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

5 \diamondsuit 4

is 8.

Substituting

x = 5

and

y = 4

into the operation

x \diamondsuit y = 3x - 2y + 1

yields

3(5) - 2(4) + 1 = 15 - 8 + 1 = 8

.

Adım Adım Çözüm

1

Substitute the operands into the custom operator rule

5 \diamondsuit 4 = 3(5) - 2(4) + 1

The first term

5

corresponds to

x

and the second term

4

corresponds to

y

in the formula.

2

Simplify the numeric expression

15 - 8 + 1 = 8

Multiply first, then perform subtraction and addition from left to right.

Anahtar Kavram

Function Definitions, Evaluation, and Custom Operators

Soru 8Soru →

For all non-zero real numbers $x$ , the function $f$ is defined by $f(x) = \frac{x^2 - 4}{2x}$ , and the function $g$ is defined by $g(x) = 3x - 1$ . What is the positive value of $a$ for which $f(g(a)) = 0$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The positive value of

a

for which

f(g(a)) = 0

is 1.

To solve

f(g(a)) = 0

, first find the values of

y

where

f(y) = 0

. The numerator of

f(y) = \frac{y^2 - 4}{2y}

equals zero when

y^2 - 4 = 0

, yielding

y = 2

and

y = -2

. Setting

g(a) = 3a - 1

equal to these roots gives

3a - 1 = 2 \implies a = 1

and

3a - 1 = -2 \implies a = -1/3

. Since

a

must be positive, the correct value is 1.

Adım Adım Çözüm

1

Determine the values of the argument

y

that satisfy

f(y) = 0

.

y = 2

or

y = -2

.

A fraction equals zero when its numerator is zero and its denominator is non-zero. Setting

y^2 - 4 = 0

gives

y = \pm 2

.

2

Substitute

g(a) = 3a - 1

into

y

to solve for

a

.

Solving

3a - 1 = 2

yields

a = 1

; solving

3a - 1 = -2

yields

a = -1/3

.

Setting the expression for

g(a)

equal to each root of

f(y) = 0

identifies all potential values for

a

.

3

Select the value of

a

matching the positivity condition.

a = 1

.

The question explicitly specifies finding the positive value of

a

.

Anahtar Kavram

Function composition and evaluation of nested functional equations.

Soru 9Soru →

For all non-zero real numbers $a$ and $b$ , the binary operator $\diamond$ is defined by $a \diamond b = a - \frac{b}{a}$ . If $k$ is a positive integer such that $(k \diamond 3) \diamond 3 = \frac{1}{2}$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive integer value of

k

is 3.

Applying the non-commutative custom operator definition

a \diamond b = a - \frac{b}{a}

to the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

requires working from the inside out. Letting

u = k \diamond 3 = k - \frac{3}{k}

, the outer equation becomes

u \diamond 3 = u - \frac{3}{u} = \frac{1}{2}

. Multiplying through by

2u

gives

2u^2 - u - 6 = 0

, which factors as

(2u + 3)(u - 2) = 0

, yielding

u = 2

or

u = -\frac{3}{2}

. Substituting

u = 2

back into

k - \frac{3}{k} = 2

produces

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Testing

u = -\frac{3}{2}

gives

2k^2 + 3k - 6 = 0

, which has no integer solutions.

Adım Adım Çözüm

1

Substitute a temporary variable for the inner custom operator expression.

Let

u = k \diamond 3 = k - \frac{3}{k}

.

Simplifies the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

to a single variable operation

u \diamond 3 = \frac{1}{2}

.

2

Apply the operator definition to

u \diamond 3

and solve the resulting quadratic equation for

u

.

u - \frac{3}{u} = \frac{1}{2} \implies 2u^2 - u - 6 = 0 \implies (2u + 3)(u - 2) = 0

, so

u = 2

or

u = -\frac{3}{2}

.

Clearing the denominator

u

transforms the custom operator equation into a standard quadratic equation.

3

Solve for positive integer

k

using the obtained values of

u

.

Setting

k - \frac{3}{k} = 2

yields

k^2 - 2k - 3 = 0 \implies (k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Setting

k - \frac{3}{k} = -\frac{3}{2}

yields

2k^2 + 3k - 6 = 0

, which has non-integer discriminant

\sqrt{57}

.

Only

u = 2

yields an integer solution for

k

, satisfying all problem constraints.

Anahtar Kavram

Custom Operator Evaluation and Nested Functional Equations

Soru 10Soru →

For all non-zero real numbers $x$ , the function $f$ satisfies the equation $2f(x) + f\left(\frac{1}{x}\right) = 6x$ . What is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

f(2)

is 7.

Substituting

x = 2

into the functional equation gives

2f(2) + f\left(\frac{1}{2}\right) = 12

. Substituting

x = \frac{1}{2}

gives

2f\left(\frac{1}{2}\right) + f(2) = 3

. Multiplying the first equation by 2 and subtracting the second equation eliminates

f\left(\frac{1}{2}\right)

, producing

3f(2) = 21

, so

f(2) = 7

.

Adım Adım Çözüm

1

Substitute

x = 2

into the functional equation

2f(2) + f\left(\frac{1}{2}\right) = 12

This establishes a linear relation between

f(2)

and

f\left(\frac{1}{2}\right)

.

2

Substitute

x = \frac{1}{2}

into the functional equation

2f\left(\frac{1}{2}\right) + f(2) = 3

This establishes a second linear relation involving the reciprocal input.

3

Eliminate

f\left(\frac{1}{2}\right)

by solving the system of equations

f(2) = 7

Multiplying the first equation by 2 yields

4f(2) + 2f\left(\frac{1}{2}\right) = 24

. Subtracting the second equation gives

3f(2) = 21

, which simplifies to

f(2) = 7

.

Anahtar Kavram

Evaluating functional equations by reciprocal variable substitution to form and solve a linear system.

Tahmini Süre:2m 0s

Soru 11Soru →

For all real numbers $x$ except $0$ and $1$ , the function $f(x)$ satisfies the relation $f(x) + 2f\left(\frac{1}{1-x}\right) = 9x$ . What is the value of $f(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

f(2)

is

6

.

Evaluating the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x

at the values

x = 2

,

x = -1

, and

x = \frac{1}{2}

creates a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this system yields

9f(2) = 54

, which simplifies directly to

f(2) = 6

.

Adım Adım Çözüm

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) \implies f(2) + 2f(-1) = 18

Applying the input

x = 2

creates an equation connecting

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) \implies f(-1) + 2f\left(\frac{1}{2}\right) = -9

Evaluating at the new input

x = -1

generates a second equation connecting

f(-1)

and

f\left(\frac{1}{2}\right)

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) \implies f\left(\frac{1}{2}\right) + 2f(2) = 4.5

Evaluating at

x = \frac{1}{2}

completes the cyclic chain by linking back to

f(2)

.

4

Solve the system of three linear equations for

f(2)

From equation 3,

f\left(\frac{1}{2}\right) = 4.5 - 2f(2)

. Substituting into equation 2 yields

f(-1) = 4f(2) - 18

. Substituting into equation 1 gives

f(2) + 2(4f(2) - 18) = 18 \implies 9f(2) = 54 \implies f(2) = 6

.

Eliminating

f\left(\frac{1}{2}\right)

and

f(-1)

isolated

f(2)

to find its exact numerical value.

Anahtar Kavram

Cyclic Functional Equations and Substitution Systems

Soru 12Soru →

For all real numbers $a$ and $b$ such that $a + b \neq 0$ , the custom operator $\star$ is defined by $a \star b = \frac{a - b}{a + b}$ . What is the value of $(3 \star 1) \star 2$ ?

-\frac{5}{3}

-\frac{3}{5}

0

\frac{3}{5}

\frac{5}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{3}{5}$

Cevap

-\frac{3}{5}

Evaluating the expression step-by-step according to parentheses yields

3 \star 1 = \frac{3-1}{3+1} = \frac{1}{2}

. Substituting

\frac{1}{2}

as the first input and

2

as the second input gives

\frac{1/2 - 2}{1/2 + 2} = \frac{-3/2}{5/2} = -\frac{3}{5}

.

Adım Adım Çözüm

1

Evaluate the inner custom operation inside parentheses:

3 \star 1

.

3 \star 1 = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2}

.

Follow the order of operations by resolving the grouped expression first using the definition

a \star b = \frac{a - b}{a + b}

with

a = 3

and

b = 1

.

2

Substitute the result

\frac{1}{2}

back into the main expression to compute

(\frac{1}{2}) \star 2

.

(\frac{1}{2}) \star 2 = \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

.

Apply the definition of the custom operator again, where the left operand is

\frac{1}{2}

and the right operand is

2

.

3

Simplify the complex fraction.

\frac{\frac{1}{2} - \frac{4}{2}}{\frac{1}{2} + \frac{4}{2}} = \frac{-\frac{3}{2}}{\frac{5}{2}} = -\frac{3}{5}.

Combine the fractions in the numerator and denominator, then divide.

Anahtar Kavram

Custom Operators and Order of Operations

Tahmini Süre:1m 30s

Soru 13Soru →

For any real numbers $u$ and $v$ , the binary operator $\star$ is defined by $u \star v = 2u^2 - 3v$ . A function $f$ is defined by $f(x) = x \star 4$ , and a function $g$ is defined by $g(x) = 3x + 1$ . If $m$ is a real number such that $f(g(m)) = 38$ , what is the product of all possible values of $m$ ?

-\frac{8}{3}

-1

-\frac{16}{9}

\frac{4}{3}

\frac{8}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{8}{3}$

Cevap

The product of all possible values of

m

is

-\frac{8}{3}

.

First, evaluate

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

. Next, substitute

g(m) = 3m + 1

into

f(x)

to get

f(g(m)) = 2(3m + 1)^2 - 12

. Setting this expression equal to

38

gives

2(3m + 1)^2 - 12 = 38

, which simplifies to

(3m + 1)^2 = 25

. Taking the square root gives two possible linear equations:

3m + 1 = 5

(which gives

m = \frac{4}{3}

) and

3m + 1 = -5

(which gives

m = -2

). Multiplying these two solutions yields

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

. Thus, the option equal to

-\frac{8}{3}

is correct.

Adım Adım Çözüm

1

Evaluate the function

f(x)

using the definition of the custom operator

\star

.

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

Substitute

u = x

and

v = 4

into the formula

u \star v = 2u^2 - 3v

.

2

Express the nested function

f(g(m))

in terms of

m

.

f(g(m)) = 2(g(m))^2 - 12 = 2(3m + 1)^2 - 12

Substitute

g(m) = 3m + 1

into

f(x)

.

3

Set

f(g(m))

equal to

38

and solve for

(3m + 1)^2

.

2(3m + 1)^2 - 12 = 38 \implies 2(3m + 1)^2 = 50 \implies (3m + 1)^2 = 25

Isolate the squared binomial term using basic algebraic manipulation.

4

Take the square root of both sides to find all possible values of

m

.

3m + 1 = 5 \implies m = \frac{4}{3}

or

3m + 1 = -5 \implies m = -2

A positive real number has both positive and negative square roots.

5

Calculate the product of the two solutions for

m

.

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

Multiply the two roots together as requested by the question stem.

Anahtar Kavram

Evaluating custom binary operators and nested composite functions, solving quadratic equations, and finding products of roots.

Soru 14Soru →

For all non-zero real numbers $x$ and $y$ , the custom operation $\star$ is defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . If $p$ and $q$ are positive numbers such that $p + q = 5$ and $p \star q = \frac{5}{6}$ , what is the value of $(p + 1) \star q$ ?

\frac{3}{2}

-\frac{3}{2}

\frac{24}{5}

0

\frac{21}{10}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{2}$

Cevap

\frac{3}{2}

By simplifying

p \star q = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

, we substitute

p + q = 5

to get

\frac{5(p - q)}{pq} = \frac{5}{6}

, which reduces to

6(p - q) = pq

. Substituting

q = 5 - p

yields the quadratic equation

p^2 + 7p - 30 = 0

. Since

p > 0

, we find

p = 3

and

q = 2

. Evaluating

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

.

Adım Adım Çözüm

1

Express the custom operation using algebraic fractions and factor the numerator

p \star q = \frac{p}{q} - \frac{q}{p} = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

Finding a common denominator converts the expression into a form where known identities can be substituted.

2

Substitute

p + q = 5

and

p \star q = \frac{5}{6}

into the simplified expression

\frac{5(p - q)}{pq} = \frac{5}{6} \implies \frac{p - q}{pq} = \frac{1}{6} \implies 6(p - q) = pq

Simplifying by dividing both sides by 5 establishes a direct relation between the difference and product of p and q.

3

Substitute

q = 5 - p

to form and solve a quadratic equation for

p

6(2p - 5) = p(5 - p) \implies 12p - 30 = 5p - p^2 \implies p^2 + 7p - 30 = 0 \implies (p + 10)(p - 3) = 0

Since p and q are positive numbers, p must be positive, giving p = 3 and q = 2.

4

Evaluate

(p + 1) \star q

with

p = 3

and

q = 2

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

Applying the custom operator definition to inputs 4 and 2 yields the final value.

Anahtar Kavram

Custom Operator Evaluation and Quadratic System Solving

Soru 15Soru →

For any real number $x$ , the function $h$ is defined by $h(x) = 3x^2 - 5x + 4$ . What is the value of $h(3)$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

To find

h(3)

, substitute

x = 3

into the function

h(x) = 3x^2 - 5x + 4

. First, compute

3^2 = 9

. Multiplying by 3 yields 27. Subtracting

5(3) = 15

gives 12, and adding 4 results in the final value of 16.

Adım Adım Çözüm

1

Substitute the value

x = 3

into the definition of

h(x)

.

h(3) = 3(3)^2 - 5(3) + 4

To evaluate a function at a specific value, replace all instances of the input variable with that value.

2

Evaluate the exponent and carry out the multiplications.

h(3) = 3(9) - 15 + 4 = 27 - 15 + 4

Follow standard order of operations: compute powers before performing multiplication.

3

Perform the remaining addition and subtraction.

h(3) = 16

Complete the calculation from left to right.

Anahtar Kavram

Function Evaluation

Soru 16Soru →

For all real numbers $x$ and $y$ , the custom operation $\Delta$ is defined by $x \Delta y = (x + 1)(y - 2)$ . If the function $g$ is defined by $g(n) = n^2 - 3$ for all real numbers $n$ , what is the value of $g(3) \Delta g(2)$ ?

Cevabı ve açıklamayı göster

Cevap: -7

Cevap

The value of

g(3) \Delta g(2)

is

-7

.

Evaluating

g(3)

gives

3^2 - 3 = 6

, and evaluating

g(2)

gives

2^2 - 3 = 1

. Substituting these values as

x = 6

and

y = 1

into the custom operator definition

x \Delta y = (x + 1)(y - 2)

yields

(6 + 1)(1 - 2) = 7 \times (-1) = -7

.

Adım Adım Çözüm

1

Evaluate the inner function

g(n)

at

n = 3

g(3) = 3^2 - 3 = 9 - 3 = 6

Before evaluating the custom operator, the inputs

g(3)

and

g(2)

must be explicitly calculated.

2

Evaluate the inner function

g(n)

at

n = 2

g(2) = 2^2 - 3 = 4 - 3 = 1

Determine the second operand for the custom operation.

3

Apply the custom operation

\Delta

to the results from Step 1 and Step 2

6 \Delta 1 = (6 + 1)(1 - 2) = 7 \times (-1) = -7

Substitute

x = 6

and

y = 1

into the formula

x \Delta y = (x + 1)(y - 2)

.

Anahtar Kavram

Evaluating functions and applying non-commutative custom operators

Soru 17Soru →

For all non-zero real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = \frac{a^2 - b^2}{ab}$ . A function $f$ is defined for all positive numbers $x$ by $f(x) = (x + 1) \star x$ . What is the value of $f(3) \star f(2)$ ?

-\frac{3}{70}

\frac{51}{70}

-\frac{51}{70}

-\frac{56}{45}

-6

Cevabı ve açıklamayı göster

Cevap: $-\frac{51}{70}$

Cevap

-\frac{51}{70}

First, evaluate the inner function values:

f(3) = 4 \star 3 = \frac{4^2 - 3^2}{4 \cdot 3} = \frac{7}{12}

and

f(2) = 3 \star 2 = \frac{3^2 - 2^2}{3 \cdot 2} = \frac{5}{6} = \frac{10}{12}

. Next, evaluate the outer operation

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{(7/12)^2 - (10/12)^2}{(7/12)(10/12)} = \frac{49 - 100}{70} = -\frac{51}{70}

. Hence, the correct response is

-\frac{51}{70}

.

Adım Adım Çözüm

1

Evaluate

f(3)

using the definition of

f(x)

and the custom operator

\star

.

f(3) = (3 + 1) \star 3 = 4 \star 3 = \frac{4^2 - 3^2}{4 \times 3} = \frac{16 - 9}{12} = \frac{7}{12}

By definition,

f(x) = (x+1) \star x

. Setting

x=3

yields

4 \star 3

.

2

Evaluate

f(2)

using the definition of

f(x)

and the custom operator

\star

.

f(2) = (2 + 1) \star 2 = 3 \star 2 = \frac{3^2 - 2^2}{3 \times 2} = \frac{9 - 4}{6} = \frac{5}{6}

Setting

x=2

in

f(x)

yields

3 \star 2

.

3

Evaluate

f(3) \star f(2) = \left(\frac{7}{12}\right) \star \left(\frac{5}{6}\right)

.

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{\left(\frac{7}{12}\right)^2 - \left(\frac{10}{12}\right)^2}{\left(\frac{7}{12}\right)\left(\frac{10}{12}\right)} = \frac{\frac{49 - 100}{144}}{\frac{70}{144}} = -\frac{51}{70}

Apply the rule

u \star v = \frac{u^2 - v^2}{uv}

with

u = \frac{7}{12}

and

v = \frac{10}{12}

.

Anahtar Kavram

Custom non-commutative binary operators and function evaluation

Tahmini Süre:2m 0s

Soru 18Soru →

For all real numbers $a$ and $b$ , the custom binary operation $\star$ is defined by $a \star b = a^2 - 2b^2 + ab$ . The function $g$ is defined by $g(x) = x \star 3$ . What is the positive value of $x$ for which $g(x) = 10$ ?

4

1

7

\sqrt{28}

3

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The positive value of

x

for which

g(x) = 10

is

4

.

Evaluating

g(x) = x \star 3

gives

x^2 - 2(3)^2 + x(3) = x^2 + 3x - 18

. Setting this expression equal to

10

produces the quadratic equation

x^2 + 3x - 28 = 0

. Factoring yields

(x + 7)(x - 4) = 0

, giving solutions

x = -7

and

x = 4

. Since

x

must be positive, the correct value is

4

.

Adım Adım Çözüm

1

Apply the definition of the custom binary operator

\star

to find

g(x)

.

g(x) = x \star 3 = x^2 - 2(3)^2 + x(3) = x^2 - 18 + 3x = x^2 + 3x - 18

Substitute

a = x

and

b = 3

into the given formula

a \star b = a^2 - 2b^2 + ab

.

2

Set the function expression equal to

10

and rearrange into standard quadratic form.

x^2 + 3x - 18 = 10 \implies x^2 + 3x - 28 = 0

Subtract

10

from both sides to form a quadratic equation equal to zero.

3

Factor the quadratic equation to find the values of

x

.

(x + 7)(x - 4) = 0 \implies x = -7 \text{ or } x = 4

Find two numbers that multiply to

-28

and add to

3

, which are

7

and

-4

.

4

Select the positive value of

x

as specified in the stem.

x = 4

The question asks specifically for the positive value of

x

.

Anahtar Kavram

Evaluating non-commutative custom operators and solving polynomial equations resulting from function evaluations.

Tahmini Süre:2m 0s

Soru 19Soru →

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = 2a - b$ . The function $f$ is defined for all real numbers by $f(x) = x^2 - 1$ . If $(x \star 3) \star f(3) = 6$ , what is the value of $x$ ?

-2

4

5

6

3

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

First, evaluate the function

f(3) = 3^2 - 1 = 8

. Next, evaluate the first custom operation

x \star 3 = 2x - 3

. Substituting these into the overall expression yields

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 14

. Setting this equal to 6 gives

4x - 14 = 6

, which simplifies to

4x = 20

, so

x = 5

.

Adım Adım Çözüm

1

Evaluate the inner function

f(3)

f(3) = 3^2 - 1 = 9 - 1 = 8

Substitute

x = 3

into the given definition

f(x) = x^2 - 1

.

2

Evaluate the inner custom operation

x \star 3

x \star 3 = 2x - 3

Apply the custom operator rule

a \star b = 2a - b

with

a = x

and

b = 3

.

3

Evaluate the outer custom operation

(x \star 3) \star f(3)

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 6 - 8 = 4x - 14

Substitute the results from Step 1 and Step 2 into the custom operator definition.

4

Solve the algebraic equation for

x

4x - 14 = 6 \implies 4x = 20 \implies x = 5

Set the simplified expression equal to 6 as given in the problem stem and isolate

x

.

Anahtar Kavram

Custom Operator Evaluation and Nested Functions

Tahmini Süre:1m 30s

Soru 20Soru →

The function

f

is defined for all real numbers by

f(x) = \begin{cases} 3x - 1 & \text{if } x \le 4 \\ x^2 - k & \text{if } x > 4 \end{cases}

where

k

is a constant. If

f(f(3)) = 14

, what is the value of

k

?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

50

To solve for

k

, evaluate the composite function

f(f(3))

step-by-step. First, find

f(3)

. Since

3 \le 4

, we use the first branch of the definition:

f(3) = 3(3) - 1 = 8

. Next, evaluate

f(8)

. Since

8 > 4

, we use the second branch of the definition:

f(8) = 8^2 - k = 64 - k

. Given that

f(f(3)) = 14

, we set

64 - k = 14

, which yields

k = 50

.

Adım Adım Çözüm

1

Evaluate the inner function value f(3)

f(3) = 8

Because 3 is less than or equal to 4, the definition f(x) = 3x - 1 applies, yielding f(3) = 3(3) - 1 = 8.

2

Express f(f(3)) in terms of k

f(8) = 64 - k

Since the inner value is 8 and 8 > 4, the piecewise condition specifies using f(x) = x^2 - k, giving f(8) = 8^2 - k = 64 - k.

3

Solve the linear equation for k

k = 50

Setting the calculated expression equal to the given value 14 gives 64 - k = 14, which simplifies to k = 50.

Anahtar Kavram

Piecewise Function Evaluation and Parameter Resolution

Tahmini Süre:1m 30s

Function Definitions, Evaluation, and Custom Operators

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