QuestionCheck if $f(x)=5x+9$ and $g(x)=\frac{x-9}{5}$ are inverse functions by finding $(f \circ g)(x)$ and $(g \circ f)(x)$ .

Studdy Solution

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.
$(g \circ f)(x) = x$

STEP 12

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 $f(x)$ .

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