For any real numbers u and v, the binary operator \star is defined by u \star v = 2u^2 - 3… — GMAT

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 $m$ ?

$-\frac{8}{3}$ Cevap
B
$-1$
C
$-\frac{16}{9}$
D
$\frac{4}{3}$
E
$\frac{8}{3}$

Cevap

The product of all possible values of

m

-\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

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

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)

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.

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

m

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

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

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

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.

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Cevap

Adım Adım Çözüm

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