For all real numbers x, the function f is defined by f(x) = x^2 - 4x + 7 and the function … — GRE General Test

For all real numbers $x$ , the function $f$ is defined by $f(x) = x^2 - 4x + 7$ and the function $g$ is defined by $g(x) = 2x + 1$ . If $f(g(k)) = 19$ and $k < 0$ , what is the value of $k$ ?

Answer: -1.5

Answer

The value of

k

-1.5

To solve for

k

, substitute

g(k) = 2k + 1

into

f(x)

to get

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

. Expanding and combining like terms yields

4k^2 - 4k + 4

. Setting this equal to

19

gives

4k^2 - 4k - 15 = 0

. Factoring the quadratic expression yields

(2k - 5)(2k + 3) = 0

, which gives

k = \frac{5}{2} = 2.5

k = -\frac{3}{2} = -1.5

. Since the problem specifies that

k < 0

, the value of

k

must be

-1.5

Step-by-Step Solution

Express the nested composite function

f(g(k))

in terms of

k

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

Substituting

g(k) = 2k + 1

for every occurrence of

x

f(x)

sets up the composite function evaluation.

Expand terms and simplify the algebraic expression

f(g(k)) = 4k^2 - 4k + 4

Expanding

(2k + 1)^2 = 4k^2 + 4k + 1

and distributing

-4(2k + 1) = -8k - 4

, then combining like terms yields

4k^2 - 4k + 4

Set the composite function expression equal to 19 and form a standard quadratic equation

4k^2 - 4k - 15 = 0

Subtracting

19

from both sides produces a standard quadratic equation

ax^2 + bx + c = 0

Solve the quadratic equation and apply the condition

k < 0

k = -1.5

Factoring

(2k - 5)(2k + 3) = 0

gives

k = 2.5

k = -1.5

. The constraint

k < 0

selects

k = -1.5

Key Concept

Composite function evaluation

f(g(x))

combined with quadratic equation solving under domain constraints.

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