QuestionFind the inverse of the function $f(x)=\frac{5}{4} x+4$ . What is $f^{-1}(x)=?$

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{5}{4}x+4$ . We are asked to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{5}{4}x +4$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \frac{5}{}y +$

STEP 4

Now, we solve this equation for $y$ to get the inverse function. First, we isolate the term with $y$ by subtracting $4$ from both sides.
$x -4 = \frac{}{4}y$

STEP 5

Finally, we divide both sides by $\frac{5}{4}$ to solve for $y$ . Remember that dividing by a fraction is the same as multiplying by its reciprocal.
$y = \frac{4}{5}(x -4)$

STEP 6

Now we replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function.
$f^{-1}(x) = \frac{4}{5}(x -4)$ So, the inverse of the function $f(x)=\frac{5}{4}x+4$ is $f^{-1}(x) = \frac{4}{5}(x -4)$ .

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QuestionFind the inverse of the function $f(x)=\frac{5}{4} x+4$ . What is $f^{-1}(x)=?$

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{5}{4}x+4$ . We are asked to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{5}{4}x +4$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \frac{5}{}y +$

STEP 4

Now, we solve this equation for $y$ to get the inverse function. First, we isolate the term with $y$ by subtracting $4$ from both sides.
$x -4 = \frac{}{4}y$

STEP 5

Finally, we divide both sides by $\frac{5}{4}$ to solve for $y$ . Remember that dividing by a fraction is the same as multiplying by its reciprocal.
$y = \frac{4}{5}(x -4)$

STEP 6

Now we replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function.
$f^{-1}(x) = \frac{4}{5}(x -4)$ So, the inverse of the function $f(x)=\frac{5}{4}x+4$ is $f^{-1}(x) = \frac{4}{5}(x -4)$ .

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QuestionFind the inverse of the function f(x)=54x+4f(x)=\frac{5}{4} x+4f(x)=45​x+4. What is f−1(x)=?f^{-1}(x)=?f−1(x)=?

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=54x+4f(x)=\frac{5}{4}x+4f(x)=45​x+4 . We are asked to find the inverse of this function, denoted as f−1(x)f^{-1}(x)f−1(x)

STEP 2

The first step in finding the inverse of a function is to replace the function notation f(x)f(x)f(x) with yyy.y=54x+4y = \frac{5}{4}x +4y=45​x+4

STEP 3

Next, we swap xxx and yyy. This means we replace every xxx in our equation with yyy and every yyy with xxx.x=5y+x = \frac{5}{}y +x=5​y+

STEP 4

Now, we solve this equation for yyy to get the inverse function. First, we isolate the term with yyy by subtracting 444 from both sides.x−4=4yx -4 = \frac{}{4}yx−4=4​y

STEP 5

Finally, we divide both sides by 54\frac{5}{4}45​ to solve for yyy. Remember that dividing by a fraction is the same as multiplying by its reciprocal.y=45(x−4)y = \frac{4}{5}(x -4)y=54​(x−4)

STEP 6

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QuestionFind the inverse of the function $f(x)=\frac{5}{4} x+4$ . What is $f^{-1}(x)=?$

Assumptions1. The function is $f(x)=\frac{5}{4}x+4$ . We are asked to find the inverse of this function, denoted as $f^{-1}(x)$

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \frac{5}{4}x +4$

Next, we swap $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \frac{5}{}y +$

Now, we solve this equation for $y$ to get the inverse function. First, we isolate the term with $y$ by subtracting $4$ from both sides.
$x -4 = \frac{}{4}y$

Finally, we divide both sides by $\frac{5}{4}$ to solve for $y$ . Remember that dividing by a fraction is the same as multiplying by its reciprocal.
$y = \frac{4}{5}(x -4)$