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Chapter 10: Problem 61

In the following exercises, find the inverse of each function. $$ f(x)=\sqrt[5]{-3 x+5} $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{5 - x^5}{3} \).

Step by step solution

01

Write the function in terms of y

Start by expressing the function as an equation with y instead of f(x). Set \( y = \root [5]{-3x + 5} \)
02

Isolate the radical term

Next, remove the fifth root by raising both sides of the equation to the power of 5. This yields \( y^5 = -3x + 5 \)
03

Solve for x

Isolate x by first subtracting 5 from both sides, followed by dividing by -3. \( y^5 - 5 = -3x \)\( x = \frac{5 - y^5}{3} \)
04

Express the inverse function

Replace y with x to express the inverse function. The inverse function is: \( f^{-1}(x) = \frac{5 - x^5}{3} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fifth Roots
In this exercise, we see a function involving a fifth root. A fifth root of a number is a value that, when multiplied by itself five times, gives the original number. For example, the fifth root of 32 is 2, because \(2^5 = 32\). Fifth roots are written using the notation \(\sqrt[5]{...}\). They are similar to square or cube roots but involve five iterations. Here, our given function is \(f(x) = \sqrt[5]{-3x+5}\). The goal is to find the inverse of this function, which involves reversing the operations applied to the input variable, x.
Isolating Variables
Isolating variables is a key step in solving equations and finding inverses of functions. In our example, we start with the equation \(y = \sqrt[5]{-3x + 5}\). The first step to isolate x is to eliminate the fifth root by raising both sides to the power of 5: this gives us \(y^5 = -3x + 5\). Next, to isolate x, we need to move all terms involving x to one side of the equation. We start by subtracting 5 from both sides, resulting in \(y^5 - 5 = -3x\). Finally, we divide both sides by -3: \( x = \frac{5 - y^5}{3} \). Now, x is isolated, and we can express the inverse function.
Solving Equations
Solving equations is the process of finding values for variables that satisfy an equation. It requires a clear understanding of inverse operations and maintaining the balance of equations. Here鈥檚 how we solve an equation step by step to find the inverse function:
  • 1. Start with the equation \(y = \sqrt[5]{-3x + 5}\).
  • 2. Raise both sides to the power of 5: \(y^5 = -3x + 5\).
  • 3. Subtract 5 from both sides: \(y^5 - 5 = -3x\).
  • 4. Divide both sides by -3: \(x = \frac{5 - y^5}{3}\).

By solving this equation for x, we write the final inverse function as \(f^{-1}(x) = \frac{5 - x^5}{3}\). This provides a clear path from the original function to its inverse.

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Most popular questions from this chapter

In the following exercises, solve each equation. \(3 \log x=\log 125\)

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Explain the method you would use to solve these equations: \(3^{x+1}=81, \quad 3^{x+1}=75 .\) Does your method require logarithms for both equations? Why or why not?

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