QuestionCheck if $f(x) = -2x + 10$ and $g(x) = \frac{x - 10}{-2}$ 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 $f(x)=-x+10$ . The function $g(x)$ is defined as $g(x)=\frac{x-10}{-}$
3. We are asked to find the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$ 4. We are asked to determine if $f(x)$ and $g(x)$ are inverse functions. Two functions are inverses of each other if and only if both $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$ .

STEP 2

First, we need to find the composition $(f \circ g)(x)$ . This is done by substituting $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f\left(\frac{x-10}{-2}\right)$

STEP 4

Now, replace $x$ in $f(x)$ with $\frac{x-10}{-2}$ .
$(f \circ g)(x) = -2\left(\frac{x-10}{-2}\right) +10$

STEP 5

implify the expression.
$(f \circ g)(x) = x -10 +10$

STEP 6

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

STEP 7

Now, we need to find the composition $(g \circ f)(x)$ . This is done by substituting $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(f(x))$

STEP 8

Substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(-2x+10)$

STEP 9

Now, replace $x$ in $g(x)$ with $-2x+$ .
$(g \circ f)(x) = \frac{-2x+-}{-2}$

STEP 10

implify the expression.
$(g \circ f)(x) = \frac{-2x}{-2}$

STEP 11

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

STEP 12

Now that we have 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 $f(x)$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms