Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Prove that the function defined by \(f(x)=a x+b\) (a linear function) for \(a \neq 0\) has an inverse function, and find \(f^{-1}(x)\). (b) Does a constant function have an inverse? Explain.

Step-by-Step Solution

Answer

(a) \( f^{-1}(x) = \frac{x - b}{a} \). (b) No, a constant function is not one-to-one.

Step 1: Understand Linear Functions

A linear function is of the form \( f(x) = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept. For the function to have an inverse, it must be bijective (one-to-one and onto). The condition \( a eq 0 \) ensures the function is one-to-one, as it creates a non-horizontal line with a constant slope.

Step 2: Verify Bijectiveness of Linear Function

Since \( a eq 0 \), every input \( x \) corresponds to a unique output \( f(x) \), making the function one-to-one. Additionally, since the line extends infinitely in both directions, the function is onto the real numbers. A function that is both one-to-one and onto is bijective, so \( f(x) \) has an inverse.

Step 3: Find the Inverse Function

To find the inverse, switch \( x \) and \( y \) in the equation \( y = ax + b \), giving us \( x = ay + b \). Solve for \( y \):\[ y = \frac{x - b}{a} \]. Thus, the inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{x - b}{a} \).

Step 4: Analyze Constant Functions

A constant function is of the form \( f(x) = c \), where \( c \) is a constant. For it to have an inverse, it must be one-to-one, meaning each output corresponds uniquely to an input. However, a constant function maps all inputs to the same output, so it is not one-to-one and does not have an inverse.

Step 5: Conclusion Based on Analysis

The linear function \( f(x) = ax + b \) with \( a eq 0 \) has an inverse, which is \( f^{-1}(x) = \frac{x - b}{a} \). In contrast, a constant function \( f(x) = c \) does not have an inverse, as it is not bijective.

Key Concepts

Linear Functions

Linear functions have a simple yet powerful form: \[ f(x) = ax + b \] where \( a \) is the slope and \( b \) is the y-intercept. These functions create straight lines when graphed. They can rise or fall with a constant rate, depending on the sign and value of \( a \).

• The slope \( a \) determines the angle and direction of the line.
• The y-intercept \( b \) is where the line crosses the y-axis.

Linear functions are vital in mathematics because they model relationships where one variable changes at a constant rate with respect to another. Such functions can be found in various real-life situations, like calculating distance over time if speed is constant.
When \( a eq 0 \), the function maintains a clear one-to-one relationship between inputs and outputs, making it eligible to have an inverse.

Bijective

A bijective function is a type of function that is both one-to-one and onto. These qualities are essential for a function to have an inverse. Let’s break it down:

• One-to-One (Injective): Each input maps to a distinct output. There are no two different inputs mapped to the same output.
• Onto (Surjective): Every possible output is covered by the function. The outputs span the entire range that is intended.

So, a bijective function efficiently uses its entire domain and range, mapping all inputs and outputs precisely without repetition or leftover.
For the linear function \( f(x) = ax + b \) where \( a eq 0 \), this bijectiveness is naturally met. Its structure ensures that each \( x \) has one unique \( f(x) \) and covers all possible real numbers, thus making an inverse possible.

Constant Function

A constant function is defined simply as: \[ f(x) = c \] where \( c \) is a constant value. Regardless of the input, the output remains \( c \).

• This results in a horizontal line on a graph.
• Commonly used to represent situations where there is no change or variation through time or conditions.

However, due to its nature, a constant function cannot have an inverse. Why? Because multiple inputs are mapped to this single constant output. In terms of functions, this means it is not one-to-one. Different inputs cannot produce the same output in a function with an inverse.
In essence, a constant function lacks the unique mapping required for each input-output pair that an inverse demands.

One-to-One Function

Understanding one-to-one functions is key for diving deeper into function inverses. In a one-to-one function, each element of the domain (input) is paired with a unique element of the range (output). No two different inputs should map to the same output.

• Imagine a line of people where each has a unique number. If one number matches two people, it's not one-to-one.
• Enables clear tracing backward: Given an output, you can trace back to exactly one input.

These functions allow us to find an inverse function since every output corresponds to a unique input.
The linear function \( f(x) = ax + b \) automatically becomes one-to-one if \( a eq 0 \). This is because the slope \( a \) ensures that no two inputs will yield the same output. Consequently, tracing backwards from any output is straightforward, helping solidify the function's invertibility.