QuestionFind the inverse function $f^{-1}(x)$ for $f(x)=\frac{1}{9} x-2$ .

Studdy Solution

STEP 1

Assumptions1. The function is given as $f(x)=\frac{1}{9} x-$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

The first step to finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{1}{9}x -2$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ with $y$ and every $y$ with $x$ .
$x = \frac{1}{9}y -2$

STEP 4

Now, we need to solve this equation for $y$ to get the inverse function. First, add2 to both sides of the equation to isolate the term with $y$ .
$x +2 = \frac{1}{9}y$

STEP 5

Next, multiply both sides of the equation by9 to solve for $y$ .
$9(x +2) = y$

STEP 6

implify the equation.
$y =9x +18$

STEP 7

Finally, replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function.
$f^{-1}(x) =9x +18$ So, the inverse of the function $f(x)=\frac{1}{9} x-2$ is $f^{-1}(x)=9x +18$ .

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QuestionFind the inverse function $f^{-1}(x)$ for $f(x)=\frac{1}{9} x-2$ .

Studdy Solution

STEP 1

Assumptions1. The function is given as $f(x)=\frac{1}{9} x-$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

The first step to finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{1}{9}x -2$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ with $y$ and every $y$ with $x$ .
$x = \frac{1}{9}y -2$

STEP 4

Now, we need to solve this equation for $y$ to get the inverse function. First, add2 to both sides of the equation to isolate the term with $y$ .
$x +2 = \frac{1}{9}y$

STEP 5

Next, multiply both sides of the equation by9 to solve for $y$ .
$9(x +2) = y$

STEP 6

implify the equation.
$y =9x +18$

STEP 7

Finally, replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function.
$f^{-1}(x) =9x +18$ So, the inverse of the function $f(x)=\frac{1}{9} x-2$ is $f^{-1}(x)=9x +18$ .

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QuestionFind the inverse function f−1(x)f^{-1}(x)f−1(x) for f(x)=19x−2f(x)=\frac{1}{9} x-2f(x)=91​x−2.

Studdy Solution

STEP 1

Assumptions1. The function is given as f(x)=19x−f(x)=\frac{1}{9} x-f(x)=91​x− . We need to find the inverse of this function, denoted as f−1(x)f^{-1}(x)f−1(x)

STEP 2

The first step to finding the inverse of a function is to replace the function notation f(x)f(x)f(x) with yyy.y=19x−2y = \frac{1}{9}x -2y=91​x−2

STEP 3

Next, we swap xxx and yyy. This means we replace every xxx with yyy and every yyy with xxx.x=19y−2x = \frac{1}{9}y -2x=91​y−2

STEP 4

Now, we need to solve this equation for yyy to get the inverse function. First, add2 to both sides of the equation to isolate the term with yyy.x+2=19yx +2 = \frac{1}{9}yx+2=91​y

STEP 5

Next, multiply both sides of the equation by9 to solve for yyy.9(x+2)=y9(x +2) = y9(x+2)=y

STEP 6

implify the equation.y=9x+18y =9x +18y=9x+18

STEP 7

Finally, replace yyy with f−1(x)f^{-1}(x)f−1(x) to denote that this is the inverse function.f−1(x)=9x+18f^{-1}(x) =9x +18f−1(x)=9x+18So, the inverse of the function f(x)=19x−2f(x)=\frac{1}{9} x-2f(x)=91​x−2 is f−1(x)=9x+18f^{-1}(x)=9x +18f−1(x)=9x+18.

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QuestionFind the inverse function $f^{-1}(x)$ for $f(x)=\frac{1}{9} x-2$ .

Assumptions1. The function is given as $f(x)=\frac{1}{9} x-$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

The first step to finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{1}{9}x -2$

Next, we swap $x$ and $y$ . This means we replace every $x$ with $y$ and every $y$ with $x$ .
$x = \frac{1}{9}y -2$

Now, we need to solve this equation for $y$ to get the inverse function. First, add2 to both sides of the equation to isolate the term with $y$ .
$x +2 = \frac{1}{9}y$

Next, multiply both sides of the equation by9 to solve for $y$ .
$9(x +2) = y$

implify the equation.
$y =9x +18$

Finally, replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function.
$f^{-1}(x) =9x +18$ So, the inverse of the function $f(x)=\frac{1}{9} x-2$ is $f^{-1}(x)=9x +18$ .