Problem 29 - Algebra A Combined Function

Question

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x} \).

Step 1: Understand the Function

The given function is \( f(x) = x^3 \). It is a one-to-one function, meaning each output from the function corresponds to exactly one input.

Step 2: Replace f(x) with y

Set \( y = x^3 \) to represent the function in terms of \( y \). This makes it easier to solve for \( x \) and find the inverse.

Step 3: Solve for x

To find the inverse, solve the equation \( y = x^3 \) for \( x \). This involves taking the cube root of both sides: \( x = \sqrt[3]{y} \).

Step 4: Express the Inverse Function

Now write the function \( x = \sqrt[3]{y} \) in inverse notation. Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x} \).

Step 5: Graph Both Functions

Graph \( f(x) = x^3 \) and \( f^{-1}(x) = \sqrt[3]{x} \) on the same set of axes. Both graphs should be symmetrical with respect to the line \( y = x \). The original function will increase through the origin, and the inverse will do the same but will be the 'flattened' version due to the cube root.

Key Concepts

One-to-One Functions

One-to-one functions are an essential concept in mathematics. They are functions where each input corresponds to a unique output, and vice versa. This means that if you have two different inputs, they will always produce two different outputs.

• This is important because it ensures that the function has an inverse.
• The inverse function reverses this process, taking outputs of the original function back to their corresponding inputs.

A visual cue that a function is one-to-one is if it passes the "horizontal line test."
If a horizontal line crosses the graph of the function at more than one point, then the function is not one-to-one. For our function, \(f(x)=x^3\), each \(x\) value maps to a unique \(y\) value, fulfilling the one-to-one condition. This property allows the inverse to be found with confidence that each function will properly correspond to its inputs and outputs.

Cube Root

The cube root is the operation that reverses cubing a number. In simpler terms, if you cube a number (multiply it by itself twice more), you use the cube root to get back to the original number. It is denoted by the symbol \(\sqrt[3]{}\).

• For the function \(f(x)=x^3\), the inverse is found by taking the cube root of \(y\).
• This gives us the inverse function \(f^{-1}(x) =\sqrt[3]{x}\).

Unlike square roots, where a number has both a positive and a negative root, the cube root always produces one real number output.
Cubing and cube rooting are also consistent across both negative and positive numbers. This means, our function and its inverse can handle negative, positive, and zero inputs equally effectively.

Graphing Functions

Graphing is a visual way to understand the behavior of functions and their inverses.
The function \(f(x) = x^3\) is a simple cubic function. Its graph is a smooth curve that passes through the origin \((0, 0)\) and extends infinitely in both upper and lower directions, steepening as it moves away from the origin.

• The graph of \(f^{-1}(x) = \sqrt[3]{x}\) is also a curve passing through the origin.
• However, this curve is less steep, resembling a "flattened" cubic graph.

Both graphs are symmetrical to each other along the line \(y=x\), because an inverse reflects the original function across this line. When graphing both functions on the same set of axes, this symmetry is a useful visual check that you have properly identified the inverse.