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

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) $( g \circ f ) ^ { - 1 } = 5 x + 13$

Quizplus · Accepted Answer

To find $( g \circ f ) ^ { - 1 }$, we need to undo the composition of the two functions $f$ and $g$. First, we need to apply $f$ to $x$ to get $f(x) = x + 4$. Then, we need to apply $g$ to $f(x)$, giving us $g(f(x)) = 5(x + 4) - 7 = 5x + 13$. Now, we want to undo this last step and solve for $x$. First, we subtract 13 from both sides to get $5x = -13 + g(f(x))$. Then, we divide both sides by 5 to get $x = \frac{1}{5}g(f(x)) - \frac{13}{5}$. This gives us the inverse function of $g \circ f$, so $( g \circ f ) ^ { - 1 } = \frac { x - 13 } { 5 }$. Therefore, the correct choice is C.

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

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