QuestionFind the inverse function of $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y=\left(\frac{\sqrt[]{x}}{4}\right)^{7}$

STEP 3

The next step in finding the inverse of a function is to switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\left(\frac{\sqrt[3]{y}}{}\right)^{7}$

STEP 4

Now, we need to solve this equation for $y$ . First, we can get rid of the7th power by taking the7th root of both sides.
$\sqrt[7]{x}=\frac{\sqrt[3]{y}}{4}$

STEP 5

Next, we can get rid of the fraction by multiplying both sides by4.
$4\sqrt[7]{x}=\sqrt[3]{y}$

STEP 6

Finally, we can get $y$ by itself by cubing both sides of the equation.
$y=(4\sqrt[]{x})^3$

STEP 7

implify the right side of the equation to get the inverse function.
$y=4^3\cdot x^{\frac{3}{7}}$ $y=64x^{\frac{3}{7}}$ So, the inverse function $f^{-1}(x)$ is $f^{-1}(x)=64x^{\frac{3}{7}}$

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QuestionFind the inverse function of $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y=\left(\frac{\sqrt[]{x}}{4}\right)^{7}$

STEP 3

The next step in finding the inverse of a function is to switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\left(\frac{\sqrt[3]{y}}{}\right)^{7}$

STEP 4

Now, we need to solve this equation for $y$ . First, we can get rid of the7th power by taking the7th root of both sides.
$\sqrt[7]{x}=\frac{\sqrt[3]{y}}{4}$

STEP 5

Next, we can get rid of the fraction by multiplying both sides by4.
$4\sqrt[7]{x}=\sqrt[3]{y}$

STEP 6

Finally, we can get $y$ by itself by cubing both sides of the equation.
$y=(4\sqrt[]{x})^3$

STEP 7

implify the right side of the equation to get the inverse function.
$y=4^3\cdot x^{\frac{3}{7}}$ $y=64x^{\frac{3}{7}}$ So, the inverse function $f^{-1}(x)$ is $f^{-1}(x)=64x^{\frac{3}{7}}$

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QuestionFind the inverse function of f(x)=(x34)7f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}f(x)=(43x​​)7.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=(x34)7f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}f(x)=(43x​​)7 . We need to find the inverse of this function, denoted as f−1(x)f^{-1}(x)f−1(x)

STEP 2

To find the inverse of a function, we first replace the function notation f(x)f(x)f(x) with yyy.y=(x4)7y=\left(\frac{\sqrt[]{x}}{4}\right)^{7}y=(4x​​)7

STEP 3

The next step in finding the inverse of a function is to switch the roles of xxx and yyy. This means we replace every xxx in our equation with yyy and every yyy with xxx.x=(y3)7x=\left(\frac{\sqrt[3]{y}}{}\right)^{7}x=(3y​​)7

STEP 4

Now, we need to solve this equation for yyy. First, we can get rid of the7th power by taking the7th root of both sides.x7=y34\sqrt[7]{x}=\frac{\sqrt[3]{y}}{4}7x​=43y​​

STEP 5

Next, we can get rid of the fraction by multiplying both sides by4.4x7=y34\sqrt[7]{x}=\sqrt[3]{y}47x​=3y​

STEP 6

Finally, we can get yyy by itself by cubing both sides of the equation.y=(4x)3y=(4\sqrt[]{x})^3y=(4x​)3

STEP 7

implify the right side of the equation to get the inverse function.y=43⋅x37y=4^3\cdot x^{\frac{3}{7}}y=43⋅x73​y=64x37y=64x^{\frac{3}{7}}y=64x73​So, the inverse function f−1(x)f^{-1}(x)f−1(x) isf−1(x)=64x37f^{-1}(x)=64x^{\frac{3}{7}}f−1(x)=64x73​

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QuestionFind the inverse function of $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ .

Assumptions1. The function is $f(x)=\left(\frac{\sqrt[3]{x}}{4}\right)^{7}$ . We need to find the inverse of this function, denoted as $f^{-1}(x)$

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y=\left(\frac{\sqrt[]{x}}{4}\right)^{7}$

The next step in finding the inverse of a function is to switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x=\left(\frac{\sqrt[3]{y}}{}\right)^{7}$

Now, we need to solve this equation for $y$ . First, we can get rid of the7th power by taking the7th root of both sides.
$\sqrt[7]{x}=\frac{\sqrt[3]{y}}{4}$

Next, we can get rid of the fraction by multiplying both sides by4.
$4\sqrt[7]{x}=\sqrt[3]{y}$

Finally, we can get $y$ by itself by cubing both sides of the equation.
$y=(4\sqrt[]{x})^3$

implify the right side of the equation to get the inverse function.
$y=4^3\cdot x^{\frac{3}{7}}$ $y=64x^{\frac{3}{7}}$ So, the inverse function $f^{-1}(x)$ is $f^{-1}(x)=64x^{\frac{3}{7}}$