QuestionFind the inverse of the one-to-one function $f(x)=\sqrt[3]{x+5}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f$ is one-to-one, meaning each output is produced by exactly one input. . We are asked to find the inverse of the function $f(x)=\sqrt[3]{x+5}$ .

STEP 2

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y = \sqrt[]{x+5}$

STEP 3

Next, we swap $x$ and $y$ to reflect the idea that the inverse function undoes the action of the original function.
$x = \sqrt[3]{y+5}$

STEP 4

Now, we need to solve this equation for $y$ to express the inverse function. To do this, we first cube both sides of the equation to eliminate the cube root.
$x^3 = (y+)$

STEP 5

Finally, we subtract5 from both sides of the equation to isolate $y$ .
$y = x^3 -5$ This is the inverse of the function $f(x)=\sqrt[3]{x+5}$ . We denote it as $f^{-1}(x)$ .
$f^{-1}(x) = x^3 -5$

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QuestionFind the inverse of the one-to-one function $f(x)=\sqrt[3]{x+5}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f$ is one-to-one, meaning each output is produced by exactly one input. . We are asked to find the inverse of the function $f(x)=\sqrt[3]{x+5}$ .

STEP 2

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y = \sqrt[]{x+5}$

STEP 3

Next, we swap $x$ and $y$ to reflect the idea that the inverse function undoes the action of the original function.
$x = \sqrt[3]{y+5}$

STEP 4

Now, we need to solve this equation for $y$ to express the inverse function. To do this, we first cube both sides of the equation to eliminate the cube root.
$x^3 = (y+)$

STEP 5

Finally, we subtract5 from both sides of the equation to isolate $y$ .
$y = x^3 -5$ This is the inverse of the function $f(x)=\sqrt[3]{x+5}$ . We denote it as $f^{-1}(x)$ .
$f^{-1}(x) = x^3 -5$

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QuestionFind the inverse of the one-to-one function f(x)=x+53f(x)=\sqrt[3]{x+5}f(x)=3x+5​.

Studdy Solution

STEP 1

Assumptions1. The function fff is one-to-one, meaning each output is produced by exactly one input. . We are asked to find the inverse of the function f(x)=x+53f(x)=\sqrt[3]{x+5}f(x)=3x+5​.

STEP 2

To find the inverse of a function, we first replace the function notation f(x)f(x)f(x) with yyy.y=x+5y = \sqrt[]{x+5}y=x+5​

STEP 3

Next, we swap xxx and yyy to reflect the idea that the inverse function undoes the action of the original function.x=y+53x = \sqrt[3]{y+5}x=3y+5​

STEP 4

Now, we need to solve this equation for yyy to express the inverse function. To do this, we first cube both sides of the equation to eliminate the cube root.x3=(y+)x^3 = (y+)x3=(y+)

STEP 5

Finally, we subtract5 from both sides of the equation to isolate yyy.y=x3−5y = x^3 -5y=x3−5This is the inverse of the function f(x)=x+53f(x)=\sqrt[3]{x+5}f(x)=3x+5​. We denote it as f−1(x)f^{-1}(x)f−1(x).f−1(x)=x3−5f^{-1}(x) = x^3 -5f−1(x)=x3−5

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QuestionFind the inverse of the one-to-one function $f(x)=\sqrt[3]{x+5}$ .

Assumptions1. The function $f$ is one-to-one, meaning each output is produced by exactly one input. . We are asked to find the inverse of the function $f(x)=\sqrt[3]{x+5}$ .

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y = \sqrt[]{x+5}$

Next, we swap $x$ and $y$ to reflect the idea that the inverse function undoes the action of the original function.
$x = \sqrt[3]{y+5}$

Now, we need to solve this equation for $y$ to express the inverse function. To do this, we first cube both sides of the equation to eliminate the cube root.
$x^3 = (y+)$

Finally, we subtract5 from both sides of the equation to isolate $y$ .
$y = x^3 -5$ This is the inverse of the function $f(x)=\sqrt[3]{x+5}$ . We denote it as $f^{-1}(x)$ .
$f^{-1}(x) = x^3 -5$