Math Snap

STEP 1

Assumptions1. The function $f(x)=5x+9$
. The function $g(x)=\frac{x-9}{5}$
3. We need to determine if $f(x)$ and $g(x)$ are inverse functions. This can be done by checking if $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$.

STEP 2

First, we will find the composition of $f$ and $g$, denoted as $(f \circ g)(x)$, which means we substitute $g(x)$ into $f(x)$.
$$(f \circ g)(x) = f(g(x))$$

STEP 3

Substitute $g(x)$ into $f(x)$.
$$(f \circ g)(x) = f\left(\frac{x-9}{5}\right)$$

STEP 4

Replace $x$ in $f(x)$ with $\frac{x-9}{}$.
$$(f \circ g)(x) =\left(\frac{x-9}{}\right) +9$$

STEP 5

implify the expression.
$$(f \circ g)(x) = x -9 +9$$

STEP 6

Further simplify the expression.
$$(f \circ g)(x) = x$$

STEP 7

Now, we will find the composition of $g$ and $f$, denoted as $(g \circ f)(x)$, which means we substitute $f(x)$ into $g(x)$.
$$(g \circ f)(x) = g(f(x))$$

STEP 8

Substitute $f(x)$ into $g(x)$.
$$(g \circ f)(x) = g(5x+)$$

STEP 9

Replace $x$ in $g(x)$ with $5x+9$.
$$(g \circ f)(x) = \frac{5x+9-9}{5}$$

STEP 10

implify the expression.
$$(g \circ f)(x) = \frac{5x}{5}$$

STEP 11

Further simplify the expression.
$$(g \circ f)(x) = x$$

Since both $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$, we can conclude that $f(x)$ and $g(x)$ are inverse functions.
Thus, $g(x)$ is the inverse function of $f(x)$.