Problem 86 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. If \(f(x)=2 x-5,\) find \(f^{-1}(x)\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = 2x - 5\) is \(f^{-1}(x) = (x+5)/2\).

Step 1: Setup the Function

First, write down the given function \(f(x) = 2x - 5\). In order to find the inverse, we need to replace \(f(x)\) with \(y\), making the equation \(y = 2x - 5\).

Step 2: Swap the Variables

The next step is to swap the variables x and y in the equation. This gives us \(x = 2y - 5\). This equation now represents the inverse function, but we still need to solve for y.

Step 3: Solve for y

In order to solve for y, first add 5 to both sides of the equation: \(x + 5 = 2y\). Then, divide both sides by 2 to isolate y: \(y = (x+5)/2\).

Step 4: Re-write the Inverse Function

The inverse function can now be written as \(f^{-1}(x) = (x+5)/2\) as a final step.

Key Concepts

Algebraic Operations

In the context of finding inverse functions, mastering algebraic operations is crucial. These operations, including addition, subtraction, multiplication, and division, are the building blocks for rearranging equations.

For example, to solve for the inverse of a function, we may need to add or subtract terms to isolate the desired variable. In the exercise given, we added 5 to both sides of the equation to isolate the term with the variable y. After that, noting that the y term was multiplied by 2, we divided the entire equation by 2 to solve for y.

Understanding these simple yet essential steps assists in manipulating equations to find the desired function or its inverse. As seen in the provided solution, by performing these operations systematically, we reached the expression of the inverse function.

Function Notation

When dealing with functions, function notation provides a way to efficiently express relationships between variables. It's a shorthand that tells us which variable is independent (usually x) and which is dependent (often represented as f(x) or y).

Function notation comes in handy when finding inverses, as seen in the exercise. Initially, we start with a function named f with x as its input: f(x). When we find the inverse, we denote it as f⁻¹(x), read as 'f-inverse of x.' The purpose of function notation here is to clearly differentiate between the original function and its inverse. This notation is essential for clarity in communication and is commonly adopted in higher-level mathematics.

Solving Equations

The process of solving equations is integral to find functions’ inverses. It involves re-arranging equations to find the value of an unknown variable. To solve the equation, we manipulate it using algebraic operations until the variable of interest stands alone on one side of the equation.

In our exercise, we took the original equation y = 2x - 5 and went through the steps of swapping the variables and then solving for y. This process required an understanding of how to correctly apply algebraic steps without altering the equation's integrity. The culmination of solving is to express the variable in its simplest form, which in the case of finding an inverse, provides us with f⁻¹(x), the inverse function notation.