Use the functions $f ( x ) = x + 4$ and $g ( x ) = 5 x - 7$ to find $( g \circ f ) ^ { - 1 }$.
A) $( g \circ f ) ^ { - 1 } = \frac { 5 x + 11 } { 4 }$
B) $( g \circ f ) ^ { - 1 } = 5 x - 42$
C) $( g \circ f ) ^ { - 1 } = \frac { x - 13 } { 5 }$
D) $( g \circ f ) ^ { - 1 } = \frac { - 7 x - 7 } { 5 }$
E) $( g \circ f ) ^ { - 1 } = 5 x + 13$

Question

Quizplus · Accepted Answer

First, we need to find the composition of the two functions:
$$(g\circ f)(x)=g(f(x))=g(x+4)=5(x+4)-7=5x+13$$
Then, we need to find the inverse of this composition. To do so, we can follow these steps:
$$y=(g\circ f)(x)=5x+13$$
Interchange $x$ and $y$:
$$x=(g\circ f)^{-1}(y)=5y+13$$
Solve for $y$:
$$y=\frac{x-13}{5}$$
Therefore, $(g\circ f)^{-1}(x)=\frac{x-13}{5}$, which is choice C.

Use the Functions f(x)=x+4f ( x ) = x + 4f(x)=x+4

Use the Functions $f ( x ) = x + 4$