Problem 52 - Intermediate Algebra

Question

Explain why every nonconstant linear function has an inverse.

Step-by-Step Solution

Answer

Every nonconstant linear function is bijective, hence it has an inverse.

Step 1: Define a Nonconstant Linear Function

A nonconstant linear function is typically expressed in the form \( f(x) = ax + b \), where \( a eq 0 \). This implies that the function has a slope (\( a \)) that is not zero, making it nonconstant. Functions like \( f(x) = 2x + 3 \) or \( f(x) = -x + 4 \) fall into this category.

Step 2: Determine the Conditions for an Inverse

For a function to have an inverse, it must be bijective, meaning it has to be both injective (one-to-one) and surjective (onto). This ensures that each output is uniquely paired with an input, allowing the function to be reversed.

Step 3: Show the Function is Injective

A function is injective if different inputs give different outputs. For the function \( f(x) = ax + b \), assume \( f(x_1) = f(x_2) \). This leads to \( ax_1 + b = ax_2 + b \). Simplifying leads to \( ax_1 = ax_2 \), and since \( a eq 0 \), we find \( x_1 = x_2 \). Thus, the function is injective.

Step 4: Show the Function is Surjective

A function is surjective if it covers all possible outputs in its range. For any real number \( y \), find \( x \) such that \( ax + b = y \). Solving for \( x \), we find \( x = \frac{y-b}{a} \), meaning that every real number \( y \) corresponds to a value of \( x \). Therefore, the function is surjective.

Step 5: Conclusion on Inverses

Since the nonconstant linear function \( f(x) = ax + b \) is both injective and surjective, it is bijective. Therefore, it has an inverse function. The inverse can be found by solving \( y = ax + b \) for \( x \), yielding \( x = \frac{y-b}{a} \), so the inverse function is \( f^{-1}(y) = \frac{y-b}{a} \).

Key Concepts

Linear Functions

A linear function can be thought of as a simple equation that forms a straight line on a graph. It is generally written in the form \( f(x) = ax + b \) where \( a \) and \( b \) are real numbers. The constant \( a \) represents the slope of the line, showing how steep the line is. The constant \( b \) shows where the line crosses the y-axis. If the slope \( a \) is zero, the function is constant, meaning that it produces the same output no matter the input. This "flat" line would not make a good candidate for an inverse because it does not change. Nonconstant linear functions, which have \( a eq 0 \), exhibit a consistent rate of change, making them more dynamic and suitable for inverses. Simply put, if you see a straight line that isn't horizontal, it likely represents a nonconstant linear function.

Bijective

A function is called bijective when it sets up a perfect "pairing" between the input values (domain) and the output values (range). This means that every element in the domain has a unique partner in the range, and vice versa. In mathematics, this pairing allows us to "reverse" or "invert" the function, thinking of the function as a perfect matchmaker.

• **Injective (One-to-One)**: Each element of the domain maps to a unique element in the range. For a nonconstant linear function, injectivity is demonstrated when different inputs yield different outputs. Imagine that if you put any two apples (x-values) in the basket, they would be distinguishable by their labeled jackets (f(x) values).


• **Surjective (Onto)**: Every element of the range is covered by the function. Surjectivity is shown when every potential outcome has a corresponding input. Think of this as making sure every jacket color is represented if jackets are made for apples.

A bijective function thus encompasses both features: injectivity and surjectivity. When it comes to linear functions, the property of being nonconstant (\( a eq 0 \)) confirms that these functions are both injective and surjective, making them bijective.

Injective and Surjective

Understanding the concepts of injective and surjective functions can be quite rewarding as they allow us to predict whether a function can be reversed.
Injective, or one-to-one functions, map each element from the domain to a unique element in the range. To visualize this, imagine a class where each student gets a unique locker. No two students share a locker, ensuring that each locker (output) maps to exactly one student (input). Nonconstant linear functions such as \( f(x) = ax + b \) are injective because their slopes guarantee unique outputs for different inputs.
Surjective, or onto functions, cover the entire range, meaning that every possible output is paired with an input. This is like making sure there is a student for each locker in the school, leaving none unassigned. With nonconstant linear functions, this property is ensured by their ability to take any real number \( y \) and find the corresponding \( x \) using the equation \( x = \frac{y-b}{a} \).
By being both injective and surjective, a linear function can have a unique inverse, meaning it can "undo" itself perfectly. This is why each nonconstant linear function is guaranteed to be bijective, leading to the existence of an inverse.