Problem 54 - College Algebra

Question

For the following exercises, find the inverse of the functions with \(a,\) c positive real numbers. $$f(x)=\sqrt[3]{a x+b}$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x^3 - b}{a} \).

Step 1: Set the function equal to y

First, set the function equal to a new variable, typically y. In this case, we write the equation as \( y = \sqrt[3]{ax + b} \). This step converts the function notation into an equation that is easier to manipulate.

Step 2: Solve for x

To solve for \( x \), we first eliminate the cube root by raising both sides of the equation to the power of 3. This gives us \( y^3 = ax + b \). Next, isolate \( x \) by subtracting \( b \) from both sides: \( y^3 - b = ax \). Finally, divide by \( a \): \( x = \frac{y^3 - b}{a} \).

Step 3: Swap variables

To find the inverse function, replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \). Thus, the equation becomes \( f^{-1}(x) = \frac{x^3 - b}{a} \).

Step 4. Confirm the inverse

Verify that the inverse function works by substituting \( f^{-1}(x) \) back into \( f(x) \) and vice versa. If \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), then your inverse is correct.

Key Concepts

Solving equations

Solving equations is a process of finding the values of variables that make an equation true. In this context, solving for the variable means isolating the variable on one side of the equation. When we have an equation like \( y = \sqrt[3]{ax+b} \), our goal is to find the value of \( x \) in terms of \( y \).

First, we eliminate the cube root by raising both sides of the equation to the power of 3. This step is known as **undoing the operation**; cube rooting and raising to the third power are opposite operations, just like adding and subtracting are opposites. So the equation \( y^3 = ax + b \) simplifies the process.

Next, we perform basic arithmetic operations: subtract **b** and then divide by **a** to solve for \( x \). This process of **isolating the variable** by using inverse operations is key in solving equations.

Cube roots

A cube root asks the question: "What number multiplied by itself three times gives this number?" Take \( \sqrt[3]{ax+b} \) from our function. Here, we want to know what number multiplied three times results in \( ax+b \).

- **Cubing** and taking the cube root are inverse operations, similar to what squaring and square rooting are to each other.- In our problem, we initially cube both sides to eliminate the cube root. This gives us the equation \( y^3 = ax+b \).

This action is like peeling back layers to find what's inside. It simplifies the equation, allowing us to isolate and solve for the desired variable, which is a critical step in determining inverse functions.

Function notation

Function notation helps us understand and solve problems involving functions. It is a concise way to represent functions, such as \( f(x) \). This often simplifies communication between mathematicians and makes equations more straightforward to work with.

- When dealing with the inverse function, we use the notation \( f^{-1}(x) \), which represents the inverse of \( f(x) \). In our exercise, to find \( f^{-1} \), we swap the roles of \( x \) and \( y \) in the function to redefine it.

In this way, function notation allows us to visualize and systematically address the relationship between input (x) and output (y), and how the concepts of function and inverse are related.

Variable substitution

Variable substitution involves replacing one variable with another or expressing variables in terms of each other. This technique is crucial for finding inverse functions.

In the exercise, after solving for \( x \) from \( y = \sqrt[3]{ax+b} \), we switch variables for substitution: swap \( y \) with \( x \) to find the inverse function. This is how we go from \( y^3 = ax+b \) to \( f^{-1}(x) = \frac{x^3-b}{a} \).

This step is essential. Not only does it allow us to express \( x \) in terms of \( y \), but it also ensures we're looking at the relationship from the inverse perspective. It shows how the original operation can be reversed to return to the initial input.