Problem 36 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. \(f(x)=3 x\) \(g(x)=\frac{1}{3} x\)

Step-by-Step Solution

Answer

Yes, these functions are inverses of each other.

Step 1: Understand the Concept of Inverse Functions

Inverse functions essentially undo each other. If two functions \(f\) and \(g\) are inverses, then both \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domains of \(g\) and \(f\), respectively.

Step 2: Calculate \(f(g(x))\)

Substitute \(g(x) = \frac{1}{3}x\) into \(f(x) = 3x\) to find \(f(g(x))\):\[f(g(x)) = f\left(\frac{1}{3}x\right) = 3 \left(\frac{1}{3}x\right) = x\]This confirms that \(f(g(x)) = x\).

Step 3: Calculate \(g(f(x))\)

Substitute \(f(x) = 3x\) into \(g(x) = \frac{1}{3}x\) to find \(g(f(x))\):\[g(f(x)) = g(3x) = \frac{1}{3}(3x) = x\]This confirms that \(g(f(x)) = x\).

Step 4: Conclude Whether \(f\) and \(g\) are Inverses

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\) are confirmed, \(f(x)=3x\) and \(g(x)=\frac{1}{3}x\) are indeed inverse functions.

Key Concepts

Function Composition

When working with functions, the concept of function composition is incredibly important. Essentially, function composition refers to applying one function to the results of another function. Let's denote two functions, \( f(x) \) and \( g(x) \). The composition of two functions is commonly represented as \( f(g(x)) \) and means that you substitute \( x \) in the function \( g \), and then apply the function \( f \) to the result.

• For the functions \( f(x) = 3x \) and \( g(x) = \frac{1}{3}x \), the composition \( f(g(x)) \) turns \( g(x) = \frac{1}{3}x \) into \( f \) by substituting: \( f\left(\frac{1}{3}x\right) = 3\left(\frac{1}{3}x\right) = x \).
• Function composition is also reversible, which means we can check the reverse composition \( g(f(x)) \) to confirm the inverse relationship. In this case, substituting \( f(x) = 3x \) into \( g \) yields \( g(3x) = \frac{1}{3}(3x) = x \).

Function composition is a crucial tool in proving whether two functions are inverse of each other.

Algebraic Functions

Algebraic functions are equations composed of variables and constants using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. The functions we're examining here, \( f(x) = 3x \) and \( g(x) = \frac{1}{3}x \), are simple linear algebraic functions.

• Linear functions in this form are particularly straightforward as inverses because they reflect simple multiplication and division, which are reversible operations.
• The key to understanding their behavior as inverses lies in symmetry through operations: multiplying by \( 3 \) and dividing by \( 3 \) are opposite operations that cancel each other out.

This symmetry is what defines these two functions as inverses, making algebraic manipulations clear and demonstrative when checking their relationships.

Mathematical Proof

Mathematical proof is a logical argument demonstrating that a certain mathematical statement is true. When verifying whether functions are inverses, we rely on proof techniques to confirm their mutual undoing effect.

• Proofs often use composition, as seen here: confirming both \( f(g(x)) = x \) and \( g(f(x)) = x \) is standard practice for proving the inverse nature of two functions.
• In the given example, detailed steps for \( f(g(x)) = x \) and \( g(f(x)) = x \) serve as direct proof of the functions' inverse relationship.

Mathematical proofs require careful consideration of each component and operation within the functions to ensure every detail is correct.

Domain of a Function

The domain of a function is the complete set of possible input values (\( x \)) for which the function is defined. The domain allows us to understand the limitations within which a function can operate without resulting in undefined or imaginary outputs.

• For the functions \( f(x) = 3x \) and \( g(x) = \frac{1}{3}x \), their domains are all real numbers because any real number can be validly substituted into either function without resulting in undefined actions or special cases.
• Recognizing a function's domain is essential in ensuring that the function compositions, such as \( f(g(x)) \) and \( g(f(x)) \), are valid across all allowable inputs.

Understanding and identifying the domain is crucial when working with functions to eliminate errors due to invalid input values.