Problem 34 - Algebra for College Students

Question

Determine whether \(f\) and \(g\) are inverse functions. $$ \begin{array}{lll} f(x)=|x-1| & \text { for } x \geq 1 & \text { and } \\ g(x)=|x+1| & \text { for } x \geq 0 & \underline{\phantom{xxx}} \end{array} $$

Step-by-Step Solution

Answer

Yes, \(f\) and \(g\) are inverse functions.

Step 1: Identify the Domain and Range of f

The function \( f(x) = |x - 1| \) is defined for \( x \geq 1 \). For this domain, \( f(x) = x - 1 \), so the range of \( f \) is \([0, \infty)\).

Step 2: Identify the Domain and Range of g

The function \( g(x) = |x + 1| \) is defined for \( x \geq 0 \). In this case, \( g(x) = x + 1 \), which has the range \([1, \infty)\).

Step 3: Consider the Composition \( f(g(x)) \)

Calculate \( f(g(x)) = f(|x + 1|) \). Since \( x \geq 0 \) for function \( g \), \( g(x) = x + 1 \geq 1 \). Therefore, \( f(g(x)) = f(x+1) = (x+1) - 1 = x \).

Step 4: Consider the Composition \( g(f(x)) \)

Calculate \( g(f(x)) = g(|x - 1|) \). Since \( x \geq 1 \) for function \( f \), \( f(x) = x - 1 \geq 0 \). Then, \( g(f(x)) = g(x-1) = (x-1) + 1 = x \).

Step 5: Verify the Inverse Relationship

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true over their respective domains, the functions \( f \) and \( g \) are confirmed to be inverses of each other.

Key Concepts

Domain and Range

Understanding the domain and range is crucial to effectively analyze functions, especially when dealing with inverse functions. The **domain** of a function is the set of all possible input values (usually represented by "x") that the function can accept without leading to any mathematical errors. In contrast, the **range** is the set of all possible output values (usually referred to as "f(x)" or "y") that result from using the domain.
Let's consider the example with the function \( f(x) = |x - 1| \) with a domain of \( x \geq 1 \). Within this range, the absolute value simplifies to \( f(x) = x - 1 \), and since \( x \) starts at 1, the resulting range is \([0, \infty)\).

• The domain of \( f \) is \( x \geq 1 \).
• The range of \( f \) is \([0, \infty)\).

Similarly, \( g(x) = |x + 1| \) is defined such that \( x \geq 0 \), making \( g(x) = x + 1 \) in that domain. Here, the starting point for "x" at 0 results in the range \([1, \infty)\).

• The domain of \( g \) is \( x \geq 0 \).
• The range of \( g \) is \([1, \infty)\).

Composition of Functions

In examining inverse functions, the composition of functions plays a vital role. Composition involves plugging one function into another to create a new function or verify certain properties. To confirm that two functions are inverses, each should return the original input when plugged into each other.Remember, in mathematical terms:

• If \( f \) and \( g \) are inverse functions, \( f(g(x)) = x \) for all \( x \) in the domain of \( g \).
• Conversely, \( g(f(x)) = x \) for all \( x \) in the domain of \( f \).

For the given functions:

• Using \( f(g(x)) = f(x+1) \), for \( x \geq 0 \), follows: \( f(x+1) = (x+1) - 1 = x \).
• Using \( g(f(x)) = g(x-1) \), for \( x \geq 1 \), follows: \( g(x-1) = (x-1) + 1 = x \).

This shows that both compositions return the input variable, satisfying the criteria for inverse functions.

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. They’re quite useful in representing real-world situations where a rule or a behavior changes after reaching a certain point.In the exercise, though not explicitly labeled as piecewise, we can view \( f(x) = |x - 1| \) to behave differently based on the input. For \( x \geq 1 \), the absolute value simplifies to \( f(x) = x - 1 \), eliminating potential negative outputs, and thus, creating a piecewise behavior.By properly understanding piecewise functions:

• We split the domain into pieces and define separate functions within these pieces.
• This approach simplifies complex functions and provides a clearer insight into behaviors like inverses and compositions.

Absolute Value Functions

Absolute value functions are essential in dealing with situations where only the magnitude of a number matters, irrespective of its sign. The absolute value of a number "x" is denoted as \(|x|\) and is always non-negative.In mathematics:

• \(|x| = x\) if \(x \geq 0\).
• \(|x| = -x\) if \(x < 0\).

In the context of the exercise, \( f(x) = |x - 1| \) and \( g(x) = |x + 1| \) are absolute value functions that modify their expressions based on their respective domains.

• For \( x \geq 1 \), \( |x - 1| \) simplifies directly to \( x - 1 \), as the inputs lead to non-negative results.
• Similarly, \( |x + 1| \) simplifies to \( x + 1 \) for \( x \geq 0 \).

Recognizing how these functions simplify aids in examining their inverses and evaluating compositions.