Question

In Exercises \(13-14\) (a) Write a function of \(x\) that performs the operations described. (b) Find the inverse and describe in words the sequence of operations in the inverse. Raise \(x\) to the fifth power, multiply by \(8,\) and then add \(4 .\)

Short answer

Answer: The sequence of operations for the inverse function is: 1. Subtract 4 from the input (\(x\)). 2. Divide the result by 8. 3. Take the fifth root of the result obtained.

Step-by-step solution

Writing the function

First, let's write the function by performing the operations given in the exercise. The function \(f(x)\) can be written as follows: $$f(x) = 8x^5 + 4$$

Finding the inverse function

To find the inverse function, we will replace \(f(x)\) with \(y\) and swap \(y\) and \(x\) in the equation, then solve for the new \(y\): $$y = 8x^5 + 4$$ Swap \(x\) and \(y\): $$x = 8y^5 + 4$$ Now solve for \(y\) to find the inverse function: Subtract \(4\) from both sides: $$x-4 = 8y^5$$ Divide both sides by \(8\): $$\frac{x-4}{8} = y^5$$ Take the fifth root of both sides: $$y = \sqrt[5]{\frac{x-4}{8}}$$ This is the inverse function, which we can write as \(f^{-1}(x) = \sqrt[5]{\frac{x-4}{8}}\).

Describing the sequence of operations in the inverse function

The inverse function is \(f^{-1}(x) = \sqrt[5]{\frac{x-4}{8}}\). The sequence of operations in words can be described as: 1. Subtract 4 from the input (\(x\)). 2. Divide the result by 8. 3. Take the fifth root of the result obtained. This sequence of operations will reverse the steps performed by the original function \(f(x)\).

Key concepts

Understanding Function Composition

In mathematics, function composition is an important concept. It involves applying one function to the results of another function. When you have two functions, say, \( f(x) \) and \( g(x) \), composing these functions means creating a new function \( h(x) \) such that \( h(x) = f(g(x)) \). This process can be visualized as connecting the output of \( g(x) \) directly into \( f(x) \) as its input.

Function composition is like a series of operations, where one set of operations leads directly into another. In our original problem, you perform a series of simpler actions such as raising to powers, multiplying, and adding.

• First, you raise \( x \) to the fifth power.
• Then, multiply the result by 8.
• Finally, add 4 to get the result of the function \( f(x) \).

This structured order is essential because each composition depends on the previous outcome. Understanding this in a clear step-by-step process helps track changes through each operation, ensuring calculations are carried out correctly.

Mastering Algebraic Manipulation

Algebraic manipulation is the process of rearranging or simplifying equations to solve for specific variables. It’s a crucial skill, especially in finding the inverse of functions. With the original function \( f(x) = 8x^5 + 4 \), algebraic manipulation helps us transform and isolate variables to uncover the inverse.

Here's how algebraic manipulation assists in solving for the inverse function:

• First, substitute \( f(x) \) with \( y \) to form the equation \( y = 8x^5 + 4 \).
• Next, swap \( x \) and \( y \) to start isolating \( y \), giving \( x = 8y^5 + 4 \).
• To isolate \( y \), subtract 4 from both sides.
• Then divide every term by 8 to simplify further.

This leads to the equation \( y^5 = \frac{x-4}{8} \). Algebraic manipulation lets us unravel complex expressions by simplifying terms step-by-step, which ultimately aids in deriving the inverse function.

Exploring Inverse Operations

Inverse operations are the backbone of solving inverse functions. The idea is to undo each operation of the original function in reverse order. For the original function \( f(x) = 8x^5 + 4 \), the goal is to reverse each operation:

1. The function first multiplies by 8 — so in the inverse, you divide by 8.
2. It was then powered by 5, which means taking the fifth root will reverse it in the inverse function.
3. Lastly, the function adds 4 — therefore, to reverse it in the inverse function, you subtract 4.

These reversed steps lead to the inverse function \( f^{-1}(x) = \sqrt[5]{\frac{x-4}{8}} \). Through inverse operations, each step nullifies the original function's effect in sequence, effectively allowing us to determine which initial value could result in a given output.