Problem 35 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. $$ \begin{array}{l}{g(x)=2 x+1} \\ {f(x)=\frac{x-1}{2}}\end{array} $$

Step-by-Step Solution

Answer

The functions \( f(x) = \frac{x-1}{2} \) and \( g(x) = 2x+1 \) are inverse functions.

Step 1: Find the Composition of f and g

To determine if two functions are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). Let's start by finding \( f(g(x)) \). Substitute \( g(x) = 2x + 1 \) into \( f(x) = \frac{x - 1}{2} \):\[ f(g(x)) = f(2x + 1) = \frac{(2x + 1) - 1}{2} \]Simplify the expression:\[ f(g(x)) = \frac{2x}{2} = x \].

Step 2: Find the Composition of g and f

Next, calculate \( g(f(x)) \) by substituting \( f(x) = \frac{x - 1}{2} \) into \( g(x) = 2x + 1 \):\[ g(f(x)) = g\left(\frac{x - 1}{2}\right) = 2\left(\frac{x - 1}{2}\right) + 1 \]Simplify the expression:\[ g(f(x)) = (x - 1) + 1 = x \].

Step 3: Verify Both Compositions are Identity Functions

We have shown that \( f(g(x)) = x \) and \( g(f(x)) = x \). Since both compositions result in \( x \), the original functions \( f(x) \) and \( g(x) \) are indeed inverses of each other.

Key Concepts

Function Composition

When dealing with functions, an essential operation is function composition. This involves plugging one function into another, similar to stacking operations. This operation helps us analyze the relationship between two functions, such as determining if they are inverse to each other.
To find the composition of two functions, say \( f \) and \( g \), we denote it as \( f(g(x)) \). This means we substitute the entire function \( g(x) \) as the input to the function \( f \). Here, the result tells us how the output of one function gets modified by another.
In the inverse function exercise, composing \( f(g(x)) \) and \( g(f(x)) \) revealed key information. The fact that both compositions resolve to the identity operation on \( x \) suggests the two functions cancel each other's effects.

Identity Function

The identity function is a fundamental concept in mathematics. It acts as a "do-nothing" operation for function composition. For a function \( h(x) \), it is an identity if applying \( h \) doesn't change the value, meaning \( h(x) = x \) for all \( x \).
To determine if two functions are inverses, check if their compositions lead to an identity function, i.e., \( f(g(x)) = x \) and \( g(f(x)) = x \). These forms indicate that both functions reverse each other's operations. This property was key in confirming that \( f(x)\) and \( g(x) \) were inverses in the exercise provided, as their compositions resulted exactly in \( x \), the identity function.
This concept is pivotal in many areas of math, as it ensures the original value remains unchanged when transformations occur in reversible pairs.

Algebraic Manipulation

Algebraic manipulation involves using basic algebra rules to simplify functions or equations. It plays a crucial role in resolving compositions and verifying function inverses.
For instance, when computing \( f(g(x)) \), a substitution is made followed by simplification using algebra. Here, \( f(2x + 1) \) was simplified to \( x \). Similarly, simplifying \( g(f(x)) \) required algebraic steps to show its equivalence to \( x \).
Without clear and effective algebraic manipulation, solving compositions to determine inverse functions would be cumbersome. Keep these techniques in mind:

• Apply arithmetic operations carefully when substituting functions.
• Simplify expressions step by step, ensuring each operation follows directly and logically.
• Verify results to ensure they meet the identity requirements for inverse functions.

Mastery in algebra enhances understanding and ensures accuracy when verifying properties like function inverses.