QuestionFind the inverse function $f^{-1}(x)$ for $f(x) = \frac{x}{9} + 1$ . What is $f^{-1}(x) = [?] x + []$ ?

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\frac{y}{9}+1$

STEP 4

Now, we solve this equation for $y$ , which will give us the inverse function $f^{-1}(x)$ . First, we subtract1 from both sides of the equation.
$x-1=\frac{y}{9}$

STEP 5

Then, we multiply both sides of the equation by9 to isolate $y$ .
$9(x-1)=y$

STEP 6

Finally, we write the inverse function $f^{-1}(x)$ in terms of $x$ .
$f^{-1}(x)=9(x-1)$ So, the inverse function of $f(x)=\frac{x}{9}+1$ is $f^{-1}(x)=9(x-1)$ .

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QuestionFind the inverse function $f^{-1}(x)$ for $f(x) = \frac{x}{9} + 1$ . What is $f^{-1}(x) = [?] x + []$ ?

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\frac{y}{9}+1$

STEP 4

Now, we solve this equation for $y$ , which will give us the inverse function $f^{-1}(x)$ . First, we subtract1 from both sides of the equation.
$x-1=\frac{y}{9}$

STEP 5

Then, we multiply both sides of the equation by9 to isolate $y$ .
$9(x-1)=y$

STEP 6

Finally, we write the inverse function $f^{-1}(x)$ in terms of $x$ .
$f^{-1}(x)=9(x-1)$ So, the inverse function of $f(x)=\frac{x}{9}+1$ is $f^{-1}(x)=9(x-1)$ .

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QuestionFind the inverse function f−1(x)f^{-1}(x)f−1(x) for f(x)=x9+1f(x) = \frac{x}{9} + 1f(x)=9x​+1. What is f−1(x)=[?]x+[]f^{-1}(x) = [?] x + []f−1(x)=[?]x+[]?

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x9+1f(x)=\frac{x}{9}+1f(x)=9x​+1. . We need to find the inverse function, denoted as f−1(x)f^{-1}(x)f−1(x).

STEP 2

The first step to find the inverse of a function is to replace the function notation f(x)f(x)f(x) with yyy.y=x9+1y=\frac{x}{9}+1y=9x​+1

STEP 3

Next, we switch the roles of xxx and yyy. This means we replace every xxx in our equation with yyy and every yyy with xxx.x=y9+1x=\frac{y}{9}+1x=9y​+1

STEP 4

Now, we solve this equation for yyy, which will give us the inverse function f−1(x)f^{-1}(x)f−1(x). First, we subtract1 from both sides of the equation.x−1=y9x-1=\frac{y}{9}x−1=9y​

STEP 5

Then, we multiply both sides of the equation by9 to isolate yyy.9(x−1)=y9(x-1)=y9(x−1)=y

STEP 6

Finally, we write the inverse function f−1(x)f^{-1}(x)f−1(x) in terms of xxx.f−1(x)=9(x−1)f^{-1}(x)=9(x-1)f−1(x)=9(x−1)So, the inverse function of f(x)=x9+1f(x)=\frac{x}{9}+1f(x)=9x​+1 is f−1(x)=9(x−1)f^{-1}(x)=9(x-1)f−1(x)=9(x−1).

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QuestionFind the inverse function $f^{-1}(x)$ for $f(x) = \frac{x}{9} + 1$ . What is $f^{-1}(x) = [?] x + []$ ?

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

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

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\frac{y}{9}+1$

Now, we solve this equation for $y$ , which will give us the inverse function $f^{-1}(x)$ . First, we subtract1 from both sides of the equation.
$x-1=\frac{y}{9}$

Then, we multiply both sides of the equation by9 to isolate $y$ .
$9(x-1)=y$

Finally, we write the inverse function $f^{-1}(x)$ in terms of $x$ .
$f^{-1}(x)=9(x-1)$ So, the inverse function of $f(x)=\frac{x}{9}+1$ is $f^{-1}(x)=9(x-1)$ .