Problem 21 - Precalculus

Question

With help from your classmates, find the inverses of the functions in Exercises \(21-24\). $$f(x)=a x+b, a \neq 0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The given function is a linear function of the form \( f(x) = ax + b \), where \( a \) is not equal to zero. This function is invertible because it's a linear function with a non-zero slope \( a \).

Step 2: Use Inverse Function Concept

The inverse of a function \( f(x) \) is another function, \( f^{-1}(x) \), that undoes the action of \( f(x) \). For linear functions, this often means swapping \( x \) and \( y \) and solving for the new \( y \).

Step 3: Swap Variables

Start by replacing \( f(x) \) with \( y \) to simplify notation: \( y = ax + b \). Swap \( x \) and \( y \) to get: \( x = ay + b \).

Step 4: Solve for y

Rearrange the equation \( x = ay + b \) to solve for \( y \):1. Subtract \( b \) from both sides: \( x - b = ay \).2. Divide both sides by \( a \): \( y = \frac{x-b}{a} \).

Step 5: Write the Inverse Function

The inverse function is \( f^{-1}(x) = \frac{x-b}{a} \). This function will transform outputs of the original function back into their respective inputs.

Key Concepts

Linear Functions

Linear functions are one of the most fundamental types of functions in algebra. A linear function is typically written in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

• \( a \) is known as the slope of the line; it determines the steepness and direction.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

These functions graph as straight lines, and their defining characteristic is that they have a constant rate of change. Because they are simple and predictable, linear functions are a key starting point for understanding more complex functions.

Function Notation

Function notation is a way to express the idea of a function in mathematical terms.It uses symbols to describe the relationship between input and output values.In the function \( f(x) = ax + b \), \( f \) is the function name, \( x \) is the input, and \( ax + b \) is the rule that assigns an output value.

• \( f(x) \) is read as "f of x" and represents the value of the function \( f \) when the input is \( x \).
• This compact notation helps us easily refer to and manipulate functions.

Function notation is not only convenient for representing functions but also crucial for operations like finding the inverse or composing functions.

Solving Equations

Solving equations is the process of finding the values of variables that make the equation true.For linear functions, solving equations often involves isolating a variable.Take the equation \( x = ay + b \), for example.

• To solve for \( y \), you'll need to rearrange the equation by performing algebraic operations such as addition, subtraction, multiplication, or division.
• In our case, subtract \( b \) from both sides and then divide by \( a \) to isolate \( y \).

Solving equations is essential in algebra and many real-world applications, as it allows us to deduce unknown values based on given information.

Invertible Functions

An invertible function is a function that can be reversed, meaning it has an inverse function.For a function to be invertible, it must be one-to-one and onto.This is often the case with linear functions like \( f(x) = ax + b \), provided \( a \) is not zero.

• A function is one-to-one if it never assigns the same value to two different inputs.
• It's onto if every possible output is covered by the function's mapping.

Finding an inverse involves swapping the roles of the input and output, then solving for the new output variable.Understanding invertible functions is critical for fields like calculus and trigonometry, where solving equations often involves working backwards to find initial inputs.