Let the functions f and g be defined by f(x) = |2x - 5| and g(x) = \sqrt{x + 7} for all re… — ACT

Let the functions $f$ and $g$ be defined by $f(x) = |2x - 5|$ and $g(x) = \sqrt{x + 7}$ for all real numbers in their respective domains. If $a$ is a real number such that $(f \circ g)(a) = 3$ , what is the sum of all possible values of $a$ ?

Cevap: 3

Cevap

The sum of all possible values of

a

3

The correct answer is

3

. The composition

(f \circ g)(a) = 3

translates to

f(g(a)) = |2\sqrt{a + 7} - 5| = 3

. This absolute value relation splits into two equations:

2\sqrt{a + 7} - 5 = 3

and

2\sqrt{a + 7} - 5 = -3

. Solving the first equation yields

\sqrt{a + 7} = 4 \Rightarrow a = 9

. Solving the second equation yields

\sqrt{a + 7} = 1 \Rightarrow a = -6

. Since both

9

and

-6

are greater than or equal to

-7

, they are within the domain of the radical function. The sum of these values is

9 + (-6) = 3

Adım Adım Çözüm

Express the composition

(f \circ g)(a)

in terms of

a

|2\sqrt{a + 7} - 5| = 3

By definition of function composition,

(f \circ g)(a) = f(g(a))

. Substituting

g(a) = \sqrt{a + 7}

into the expression for

f(x)

yields

f(g(a)) = |2g(a) - 5| = |2\sqrt{a + 7} - 5|

Set up the two algebraic cases to eliminate the absolute value.

2\sqrt{a + 7} - 5 = 3

2\sqrt{a + 7} - 5 = -3

An absolute value equation of the form

|u| = c

(where

c \geq 0

) has two possible cases:

u = c

u = -c

Solve the first equation case for

a

a = 9

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = 3

gives

2\sqrt{a + 7} = 8

. Dividing by

2

gives

\sqrt{a + 7} = 4

. Squaring both sides yields

a + 7 = 16

, which gives

a = 9

Solve the second equation case for

a

a = -6

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = -3

gives

2\sqrt{a + 7} = 2

. Dividing by

2

gives

\sqrt{a + 7} = 1

. Squaring both sides yields

a + 7 = 1

, which gives

a = -6

Verify domain constraints and sum the valid solutions.

3

Both

a = 9

and

a = -6

satisfy the domain requirement for

g(x) = \sqrt{x+7}

, which is

x \geq -7

. The sum of these two valid values is

9 + (-6) = 3

Anahtar Kavram

Evaluating and solving equations containing composite functions, absolute values, and radical functions.

Alternatif Yöntem

Instead of expanding the composition immediately, substitute a temporary variable

u = g(a) = \sqrt{a+7}

. The equation becomes

f(u) = 3 \Rightarrow |2u - 5| = 3

. Solve this simplified absolute value equation to get

2u - 5 = 3 \Rightarrow u = 4

, and

2u - 5 = -3 \Rightarrow u = 1

. Next, substitute back

\sqrt{a+7}

for

u

: solving

\sqrt{a+7} = 4

yields

a = 9

, and solving

\sqrt{a+7} = 1

yields

a = -6

. Summing these two solutions gives

9 + (-6) = 3

Tahmini Süre:2m 0s

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