Problem 53 - Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. $$ f(x)=\frac{x^{7}}{2} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = (2x)^{1/7} \).

Step 1: Understand the Function

The given function is \( f(x) = \frac{x^7}{2} \). This is a power function divided by a constant.

Step 2: Replace \( f(x) \) with \( y \)

We start by replacing \( f(x) \) with \( y \) to rewrite the function as an equation. This gives us \( y = \frac{x^7}{2} \).

Step 3: Solve for \( x \) in terms of \( y \)

Begin by isolating \( x^7 \) by multiplying both sides of the equation by 2: \( 2y = x^7 \). Next, solve for \( x \) by taking the seventh root of both sides: \( x = (2y)^{1/7} \).

Step 4: Express the Inverse Function

To find the inverse function, swap \( x \) and \( y \). So, \( y = (2x)^{1/7} \) translates into \( f^{-1}(x) = (2x)^{1/7} \).

Key Concepts

One-to-One Functions

A function is described as **one-to-one** if every element of the domain maps to a unique element in the codomain. This means that for any distinct elements \( x_1 \) and \( x_2 \) in the domain, the corresponding function values \( f(x_1) \) and \( f(x_2) \) are also distinct. Essentially, different inputs lead to different outputs.

One of the main characteristics of one-to-one functions is their invertibility. If a function is one-to-one, it can be reversed, meaning each output value of the function corresponds to exactly one input value, allowing us to find its inverse. This property is essential when dealing with inverse functions, like in the given exercise.

To check if a function is one-to-one, you can use the horizontal line test on its graph. If no horizontal line intersects the graph at more than one point, the function is one-to-one.

In our example, since the given function is one-to-one, we can confidently find its inverse and express it using the inverse function notation \( f^{-1}(x) \), ensuring that each \( x \) in the inverse is mapped uniquely in the original function.

Power Functions

Power functions are mathematical expressions of the form \( f(x) = ax^n \), where \( a \) and \( n \) are constants. In these functions, \( x \) is raised to the power of \( n \), indicating the growth rate of the function.

In the exercise, the function \( f(x) = \frac{x^7}{2} \) is a type of power function where \( n=7 \). Here, the variable \( x \) is raised to the 7th power, and the coefficient is \( \frac{1}{2} \). The behavior of a power function largely depends on the exponent:

• If \( n \) is positive and even, the graph is a parabola opening upwards.
• If \( n \) is positive and odd, the graph is typically a wave-like curve.

Understanding power functions is crucial in solving for the inverse. In our problem, isolating \( x \) involves taking the seventh root, which reflects a symmetrical operation to raising \( x \) to the 7th power. Thus, recognizing power functions helps us determine the steps needed to express inverse functions accurately.

Function Notation

**Function notation** is a way of representing functions and is used to convey key information about their behavior. It mainly involves substituting the input \( (x) \) into a function to see what it outputs \( (f(x)) \). This notation not only makes equations more readable but also communicates that functions represent relationships between variables.

In our scenario, the function is initially given as \( f(x) = \frac{x^7}{2} \). Function notation tells us that for every input \( x \), we divide \( x^7 \) by 2 to get the output. When finding the inverse, our goal is to swap the roles of \( x \) and \( f(x) \) (or \( y \) when rewritten).

Expressing the inverse as \( f^{-1}(x) \) means our new function takes the value we normally consider an output, such as \( x \), and transforms it back to the original input using the inverse operations. Here, our inverse function is \( f^{-1}(x) = (2x)^{1/7} \). This notation is powerful as it succinctly indicates the reverse transformation of the original function, maintaining the elegant clarity of function relationships.