Let the functions f and g be defined by f(x) = |x - 4| and g(x) = 2x + 1 for all real numb… — ACT

Let the functions $f$ and $g$ be defined by $f(x) = |x - 4|$ and $g(x) = 2x + 1$ for all real numbers. For what values of $x$ does the composition $f(g(x)) = 5$ ?

$x = -1$ and $x = 4$ Answer
B
$x = 4$ only
C
$x = 2$ and $x = 6$
D
$x = -2$ and $x = 2$
E
$x = -4$ and $x = 4$

Answer

x = -1

and

x = 4

To find the values of

x

for which

f(g(x)) = 5

, we substitute the expression for

g(x)

into

f(x)

, yielding

|(2x + 1) - 4| = 5

, which simplifies to

|2x - 3| = 5

. Setting up the two possible cases for the absolute value gives

2x - 3 = 5

(which solves to

x = 4

) and

2x - 3 = -5

(which solves to

x = -1

). Both values of

x

satisfy the original composition equation.

Step-by-Step Solution

Substitute the inner function

g(x) = 2x + 1

into the outer function

f(x) = |x - 4|

to find the composite function

f(g(x))

f(g(x)) = |(2x + 1) - 4| = |2x - 3|

Function composition requires evaluating the outer function at the output of the inner function.

Set the composite expression equal to

5

|2x - 3| = 5

We are solving for the values of

x

where the output of the composite function is

5

Split the absolute value equation into its positive and negative cases and solve each linear equation for

x

Case 1:

2x - 3 = 5 \implies 2x = 8 \implies x = 4

. Case 2:

2x - 3 = -5 \implies 2x = -2 \implies x = -1

An absolute value equation

|u| = c

(with

c > 0

) is equivalent to

u = c

u = -c

Key Concept

Evaluating a composite function with an absolute value and solving the resulting equations by considering both positive and negative cases.

Estimated Time:1m 30s

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