QuestionCheck if $f(x)=x+0$ and $g(x)=x+18$ are inverses by finding $(f \circ g)(x)$ and $(g \circ f)(x)$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=x+0$ . The function $g(x)$ is defined as $g(x)=x+18$
3. We need to determine if $f(x)$ and $g(x)$ are inverse functions by using function composition.

STEP 2

First, we need to understand what it means for two functions to be inverses of each other. If $f$ and $g$ are inverse functions, then the composition of $f$ and $g$ in both orders should return the original input. Mathematically, this means that $f(g(x)) = x \quad \text{and} \quad g(f(x)) = x$

STEP 3

Let's first compute the composition $(f \circ g)(x)$ , which means we apply the function $f$ to the output of $g(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 4

Substitute $g(x)$ into the function $f(x)$ .
$(f \circ g)(x) = f(x+18)$

STEP 5

Substitute $x+18$ into $f(x)$ .
$(f \circ g)(x) = (x+18) +0$

STEP 6

implify the expression.
$(f \circ g)(x) = x+18$

STEP 7

Now, let's compute the composition $(g \circ f)(x)$ , which means we apply the function $g$ to the output of $f(x)$ .
$(g \circ f)(x) = g(f(x))$

STEP 8

Substitute $f(x)$ into the function $g(x)$ .
$(g \circ f)(x) = g(x+0)$

STEP 9

Substitute $x+$ into $g(x)$ .
$(g \circ f)(x) = (x+) +18$

STEP 10

implify the expression.
$(g \circ f)(x) = x+18$

STEP 11

Now, we compare the results of $(f \circ g)(x)$ and $(g \circ f)(x)$ with $x$ . Neither of them equals to $x$ . Therefore, $f(x)$ and $g(x)$ are not inverse functions.
Thus, $g(x)$ is not the inverse function of $f(x)$ .

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