Math / Algebra

QuestionFind a formula for the inverse of the following function, if possible. $V(x)=4 \sqrt[5]{x}+5$

Studdy Solution

STEP 1

What is this asking? We're trying to find a function that "undoes" $V(x)$ , meaning if we plug in $V(x)$ into this new function, we should get $x$ back! Watch out! Not all functions have inverses, so we need to make sure this one does!
Also, remember the order of operations when undoing the function.

STEP 2

1. Check for Invertibility
2. Swap and Solve

STEP 3

First, let's check if $V(x)$ is invertible.
Since $V(x)$ involves taking the fifth root and adding a constant, it's a one-to-one function, meaning it passes the horizontal line test!
This is great news because it means it does have an inverse.
Let's proceed!

STEP 4

To find the inverse, we'll swap $x$ and $y$ in our original equation.
Remember, $V(x)$ is just a fancy way of writing $y$ , so we have $y = 4 \sqrt[5]{x} + 5$ .
Swapping gives us $x = 4 \sqrt[5]{y} + 5$ .

STEP 5

Now, let's solve for $y$ .
First, we want to isolate the term with $y$ , which is $4 \sqrt[5]{y}$ .
We can do this by subtracting 5 from both sides of the equation: $x - 5 = 4 \sqrt[5]{y}$ .
Remember, we're adding $-5$ to both sides to eliminate the $5$ on the right side.

STEP 6

Next, we want to isolate $y$ by getting rid of the coefficient $4$ .
We can do this by dividing both sides of the equation by $4$ : $\frac{x - 5}{4} = \sqrt[5]{y}$ .
We're multiplying both sides by $\frac{1}{4}$ to turn the $4$ on the right side into $1$ .

STEP 7

Finally, we need to undo the fifth root to get $y$ by itself.
We can do this by raising both sides of the equation to the fifth power: $\left(\frac{x - 5}{4}\right)^5 = \left(\sqrt[5]{y}\right)^5$ which simplifies to $\left(\frac{x - 5}{4}\right)^5 = y.$

STEP 8

So, our inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ !

STEP 9

The inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ .

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QuestionFind a formula for the inverse of the following function, if possible. $V(x)=4 \sqrt[5]{x}+5$

Studdy Solution

STEP 1

STEP 2

1. Check for Invertibility
2. Swap and Solve

STEP 3

First, let's check if $V(x)$ is invertible.
Since $V(x)$ involves taking the fifth root and adding a constant, it's a one-to-one function, meaning it passes the horizontal line test!
This is great news because it means it does have an inverse.
Let's proceed!

STEP 4

To find the inverse, we'll swap $x$ and $y$ in our original equation.
Remember, $V(x)$ is just a fancy way of writing $y$ , so we have $y = 4 \sqrt[5]{x} + 5$ .
Swapping gives us $x = 4 \sqrt[5]{y} + 5$ .

STEP 5

Now, let's solve for $y$ .
First, we want to isolate the term with $y$ , which is $4 \sqrt[5]{y}$ .
We can do this by subtracting 5 from both sides of the equation: $x - 5 = 4 \sqrt[5]{y}$ .
Remember, we're adding $-5$ to both sides to eliminate the $5$ on the right side.

STEP 6

Next, we want to isolate $y$ by getting rid of the coefficient $4$ .
We can do this by dividing both sides of the equation by $4$ : $\frac{x - 5}{4} = \sqrt[5]{y}$ .
We're multiplying both sides by $\frac{1}{4}$ to turn the $4$ on the right side into $1$ .

STEP 7

Finally, we need to undo the fifth root to get $y$ by itself.
We can do this by raising both sides of the equation to the fifth power: $\left(\frac{x - 5}{4}\right)^5 = \left(\sqrt[5]{y}\right)^5$ which simplifies to $\left(\frac{x - 5}{4}\right)^5 = y.$

STEP 8

So, our inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ !

STEP 9

The inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ .

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QuestionFind a formula for the inverse of the following function, if possible. V(x)=4x5+5V(x)=4 \sqrt[5]{x}+5V(x)=45x​+5

Studdy Solution

STEP 1

STEP 2

1. Check for Invertibility2. Swap and Solve

STEP 3

First, let's **check if V(x)V(x)V(x) is invertible**.Since V(x)V(x)V(x) involves taking the fifth root and adding a constant, it's a one-to-one function, meaning it passes the horizontal line test!This is great news because it means it *does* have an inverse.Let's proceed!

STEP 4

To find the inverse, we'll **swap** xxx and yyy in our original equation.Remember, V(x)V(x)V(x) is just a fancy way of writing yyy, so we have y=4x5+5y = 4 \sqrt[5]{x} + 5y=45x​+5.Swapping gives us x=4y5+5x = 4 \sqrt[5]{y} + 5x=45y​+5.

STEP 5

STEP 6

Next, we want to **isolate** yyy by getting rid of the coefficient 444.We can do this by dividing both sides of the equation by 444: x−54=y5\frac{x - 5}{4} = \sqrt[5]{y}4x−5​=5y​.We're multiplying both sides by 14\frac{1}{4}41​ to turn the 444 on the right side into 111.

STEP 7

STEP 8

So, our **inverse function** is V−1(x)=(x−54)5V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5V−1(x)=(4x−5​)5!

STEP 9

The inverse function is V−1(x)=(x−54)5V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5V−1(x)=(4x−5​)5.

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QuestionFind a formula for the inverse of the following function, if possible. $V(x)=4 \sqrt[5]{x}+5$

1. Check for Invertibility
2. Swap and Solve

First, let's check if $V(x)$ is invertible.
Since $V(x)$ involves taking the fifth root and adding a constant, it's a one-to-one function, meaning it passes the horizontal line test!
This is great news because it means it does have an inverse.
Let's proceed!

To find the inverse, we'll swap $x$ and $y$ in our original equation.
Remember, $V(x)$ is just a fancy way of writing $y$ , so we have $y = 4 \sqrt[5]{x} + 5$ .
Swapping gives us $x = 4 \sqrt[5]{y} + 5$ .

Next, we want to isolate $y$ by getting rid of the coefficient $4$ .
We can do this by dividing both sides of the equation by $4$ : $\frac{x - 5}{4} = \sqrt[5]{y}$ .
We're multiplying both sides by $\frac{1}{4}$ to turn the $4$ on the right side into $1$ .

So, our inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ !

The inverse function is $V^{-1}(x) = \left(\frac{x - 5}{4}\right)^5$ .