Math / Algebra

QuestionInverse?
21. a) $y=\sqrt[3]{x}$

Studdy Solution

STEP 1

What is this asking? We're on a mission to find the inverse of the function $y = \sqrt[3]{x}$ , which means we want to find a new function that "undoes" what this one does! Watch out! Don't mix up inverse functions with reciprocal functions!
The reciprocal of $\sqrt[3]{x}$ is $\frac{1}{\sqrt[3]{x}}$ , which is totally different from the inverse!

STEP 2

1. Swap $x$ and $y$
2. Solve for $y$

STEP 3

To start finding the inverse, we swap $x$ and $y$ in the original equation.
This gives us a new equation: $x = \sqrt[3]{y}$ Why do we swap?
Think of it like this: the original function takes an input $x$ and gives you an output $y$ .
The inverse function does the opposite; it takes the output $y$ (which we now call $x$ ) and gives you back the original input (which we now need to solve for as the new $y$ ).

STEP 4

Now, we want to get $y$ all by itself.
Since $y$ is trapped inside a cube root, we can cube both sides of the equation to free it! $x^3 = (\sqrt[3]{y})^3$

STEP 5

Cubing a cube root is like hitting the "undo" button!
They cancel each other out, leaving us with: $x^3 = y$

STEP 6

We typically write functions with $y$ on the left side, so let's just rewrite this as: $y = x^3$ And there you have it!
This is our inverse function!

STEP 7

The inverse of the function $y = \sqrt[3]{x}$ is $y = x^3$ .

Was this helpful?

QuestionInverse?
21. a) $y=\sqrt[3]{x}$

Studdy Solution

STEP 1

STEP 2

1. Swap $x$ and $y$
2. Solve for $y$

STEP 3

STEP 4

Now, we want to get $y$ all by itself.
Since $y$ is trapped inside a cube root, we can cube both sides of the equation to free it! $x^3 = (\sqrt[3]{y})^3$

STEP 5

Cubing a cube root is like hitting the "undo" button!
They cancel each other out, leaving us with: $x^3 = y$

STEP 6

We typically write functions with $y$ on the left side, so let's just rewrite this as: $y = x^3$ And there you have it!
This is our inverse function!

STEP 7

The inverse of the function $y = \sqrt[3]{x}$ is $y = x^3$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionInverse?21. a) y=x3y=\sqrt[3]{x}y=3x​

Studdy Solution

STEP 1

STEP 2

1. Swap xxx and yyy2. Solve for yyy

STEP 3

STEP 4

Now, we want to get yyy all by itself.Since yyy is trapped inside a cube root, we can **cube both sides** of the equation to free it! x3=(y3)3x^3 = (\sqrt[3]{y})^3x3=(3y​)3

STEP 5

Cubing a cube root is like hitting the "undo" button!They cancel each other out, leaving us with: x3=yx^3 = yx3=y

STEP 6

We typically write functions with yyy on the left side, so let's just **rewrite** this as: y=x3y = x^3y=x3 And there you have it!This is our inverse function!

STEP 7

The inverse of the function y=x3y = \sqrt[3]{x}y=3x​ is y=x3y = x^3y=x3.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionInverse?
21. a) $y=\sqrt[3]{x}$

1. Swap $x$ and $y$
2. Solve for $y$

Now, we want to get $y$ all by itself.
Since $y$ is trapped inside a cube root, we can cube both sides of the equation to free it! $x^3 = (\sqrt[3]{y})^3$

Cubing a cube root is like hitting the "undo" button!
They cancel each other out, leaving us with: $x^3 = y$

We typically write functions with $y$ on the left side, so let's just rewrite this as: $y = x^3$ And there you have it!
This is our inverse function!

The inverse of the function $y = \sqrt[3]{x}$ is $y = x^3$ .