Math / Algebra

QuestionFind a formula for the inverse of the following function, if possible. $s(x)=\sqrt[3]{4 x-5}$
Answer How to enter your answer (opens in new window)
Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used. $s^{-1}(x)=$ does not have an inverse function

Studdy Solution

STEP 1

What is this asking? We're on a mission to find the inverse function of a cube root function, which means we want to discover a function that undoes the original function's actions! Watch out! Don't forget that not every function has an inverse.
A function has an inverse if and only if it's a one-to-one function.
Luckily, cube root functions are one-to-one!

STEP 2

1. Set up the inverse function.
2. Solve for the inverse.

STEP 3

Let's start by replacing $s(x)$ with $y$ to make things a bit easier to work with.
So, we have: $y = \sqrt[3]{4x - 5}$ This is just a cosmetic change to make our calculations easier to read.

STEP 4

To find the inverse, we swap $x$ and $y$ .
This gives us: $x = \sqrt[3]{4y - 5}$ This swap is the core action of finding an inverse function.
It reflects the fact that the input of the inverse is the output of the original, and vice-versa.

STEP 5

Now, we want to solve for $y$ .
Let's cube both sides of the equation to get rid of the cube root: $x^3 = (\sqrt[3]{4y - 5})^3$ $x^3 = 4y - 5$ Cubing both sides is the inverse operation of taking the cube root, and it allows us to isolate the expression inside the cube root.

STEP 6

We're almost there! Add $\textbf{5}$ to both sides: $x^3 + 5 = 4y$ Now, divide both sides by $\textbf{4}$ to isolate $y$ : $\frac{x^3 + 5}{4} = y$

STEP 7

Finally, we rewrite this as the inverse function: $s^{-1}(x) = \frac{x^3 + 5}{4}$ This is our final inverse function!
It takes an input $x$ , cubes it, adds 5, and then divides the result by 4.

STEP 8

The inverse function is: $s^{-1}(x) = \frac{x^3 + 5}{4}$

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Set up the inverse function.
2. Solve for the inverse.

STEP 3

Let's start by replacing $s(x)$ with $y$ to make things a bit easier to work with.
So, we have: $y = \sqrt[3]{4x - 5}$ This is just a cosmetic change to make our calculations easier to read.

STEP 4

To find the inverse, we swap $x$ and $y$ .
This gives us: $x = \sqrt[3]{4y - 5}$ This swap is the core action of finding an inverse function.
It reflects the fact that the input of the inverse is the output of the original, and vice-versa.

STEP 5

Now, we want to solve for $y$ .
Let's cube both sides of the equation to get rid of the cube root: $x^3 = (\sqrt[3]{4y - 5})^3$ $x^3 = 4y - 5$ Cubing both sides is the inverse operation of taking the cube root, and it allows us to isolate the expression inside the cube root.

STEP 6

We're almost there! Add $\textbf{5}$ to both sides: $x^3 + 5 = 4y$ Now, divide both sides by $\textbf{4}$ to isolate $y$ : $\frac{x^3 + 5}{4} = y$

STEP 7

Finally, we rewrite this as the inverse function: $s^{-1}(x) = \frac{x^3 + 5}{4}$ This is our final inverse function!
It takes an input $x$ , cubes it, adds 5, and then divides the result by 4.

STEP 8

The inverse function is: $s^{-1}(x) = \frac{x^3 + 5}{4}$

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.

Studdy Solution

STEP 1

STEP 2

1. Set up the inverse function.2. Solve for the inverse.

STEP 3

Let's start by replacing s(x)s(x)s(x) with yyy to make things a bit easier to work with.So, we have: y=4x−53y = \sqrt[3]{4x - 5}y=34x−5​ This is just a cosmetic change to make our calculations easier to read.

STEP 4

To find the inverse, we **swap** xxx and yyy.This gives us: x=4y−53x = \sqrt[3]{4y - 5}x=34y−5​ This swap is the core action of finding an inverse function.It reflects the fact that the input of the inverse is the output of the original, and vice-versa.

STEP 5

STEP 6

We're almost there! **Add** 5\textbf{5}5 to both sides: x3+5=4yx^3 + 5 = 4yx3+5=4y Now, **divide** both sides by 4\textbf{4}4 to isolate yyy: x3+54=y\frac{x^3 + 5}{4} = y4x3+5​=y

STEP 7

Finally, we rewrite this as the inverse function: s−1(x)=x3+54s^{-1}(x) = \frac{x^3 + 5}{4}s−1(x)=4x3+5​ This is our final inverse function!It takes an input xxx, cubes it, adds 5, and then divides the result by 4.

STEP 8

The inverse function is: s−1(x)=x3+54s^{-1}(x) = \frac{x^3 + 5}{4}s−1(x)=4x3+5​

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.

1. Set up the inverse function.
2. Solve for the inverse.

Let's start by replacing $s(x)$ with $y$ to make things a bit easier to work with.
So, we have: $y = \sqrt[3]{4x - 5}$ This is just a cosmetic change to make our calculations easier to read.

To find the inverse, we swap $x$ and $y$ .
This gives us: $x = \sqrt[3]{4y - 5}$ This swap is the core action of finding an inverse function.
It reflects the fact that the input of the inverse is the output of the original, and vice-versa.

We're almost there! Add $\textbf{5}$ to both sides: $x^3 + 5 = 4y$ Now, divide both sides by $\textbf{4}$ to isolate $y$ : $\frac{x^3 + 5}{4} = y$

Finally, we rewrite this as the inverse function: $s^{-1}(x) = \frac{x^3 + 5}{4}$ This is our final inverse function!
It takes an input $x$ , cubes it, adds 5, and then divides the result by 4.

The inverse function is: $s^{-1}(x) = \frac{x^3 + 5}{4}$