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Chapter 10: Problem 58

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

Short Answer

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

Step by step solution

01

- Replace f(x) with y

Start by replacing the function notation \( f(x) \) with \( y \). This gives us: \[ y = \sqrt[3]{x+5} \]
02

- Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This changes the equation to: \[ x = \sqrt[3]{y+5} \]
03

- Isolate y

Solve for \( y \) by performing inverse operations. First, cube both sides to eliminate the cube root: \[ x^3 = y + 5 \]. Then, subtract 5 from both sides: \[ y = x^3 - 5 \]
04

- Replace y with f^(-1)(x)

Finally, replace \( y \) with the inverse function notation \( f^{-1}(x) \). Thus, we get: \[ f^{-1}(x) = x^3 - 5 \]

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Key Concepts

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

Cube Roots
Cube roots help us find a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). In our exercise, we used the cube root to define the function \(f(x) = \sqrt[3]{x + 5}\). Knowing how to handle cube roots is essential because they are the key to undoing the cube operation, which is crucial when finding the inverse of the function.
Function Notation
Function notation is a way to name functions for easy reference. Instead of writing long equations repeatedly, we use simple notations like \(f(x)\) for functions. In the exercise, \(f(x) = \sqrt[3]{x + 5}\)\ means that \(f\) is a function defined in terms of \(x\). To find the inverse, we started by replacing \(f(x)\) with \(y\), for clarity in our algebraic manipulations. The final notation for the inverse function became \(f^{-1}(x) = x^3 - 5\). This is how we communicate that we're now talking about the inverse function.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations using algebra rules. In this exercise, we swapped \(x\) and \(y\) to find the inverse function. This means we had \(x = \sqrt[3]{y + 5}\). To isolate \(y\), first, we cubed both sides, giving \(x^3 = y + 5\). Cubing is the opposite operation of taking the cube root. Next, we solved for \(y\) by subtracting 5 from both sides to get \(y = x^3 - 5\). Finally, we replaced \(y\) with \(f^{-1}(x)\) to represent the inverse function.
Solving Equations
Solving equations means finding what values make the equation true. In the context of finding an inverse function, we go through several steps:
  • First, replace the function notation with \(y\): \(y = \sqrt[3]{x + 5}\).
  • Next, swap \(x\) and \(y\): \(x = \sqrt[3]{y + 5}\).
  • Then, use algebraic manipulations to isolate \(y\): cube both sides \(x^3 = y + 5\), and then solve for \(y\): \(y = x^3 - 5\).
  • Finally, switch back to function notation: \(f^{-1}(x) = x^3 - 5\).
By systematically applying these steps, we can solve equations to find the inverse function. Remember, solving equations often involves performing inverse operations like squaring or cubing to undo roots.

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Most popular questions from this chapter

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\frac{1}{4} e^{x}=3\)

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