Problem 56 - Elementary and Intermediate Algebra

Question

Show that each pair of functions are inverses. \(f(x)=5 x-1, f^{-1}(x)=\frac{x+1}{5}\)

Step-by-Step Solution

Answer

The functions are inverses as both compositions return the input \(x\).

Step 1: Understand the Definition of Inverse Functions

Two functions, say \(f(x)\) and \(g(x)\), are inverses if their compositions \(f(g(x))\) and \(g(f(x))\) both return the input \(x\). This means we need to show both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 2: Verify \(f(f^{-1}(x)) = x\)

First, substitute \(f^{-1}(x)\) into \(f(x)\):\[ f(f^{-1}(x)) = f\left(\frac{x+1}{5}\right) \]Now replace \(x\) in \(f(x) = 5x - 1\) with \(\frac{x+1}{5}\):\[ f(f^{-1}(x)) = 5 \left(\frac{x+1}{5}\right) - 1 \]Simplify:\[ = x + 1 - 1 \]\[ = x \]Thus, \(f(f^{-1}(x)) = x\) is proven.

Step 3: Verify \(f^{-1}(f(x)) = x\)

Next, substitute \(f(x)\) into \(f^{-1}(x)\):\[ f^{-1}(f(x)) = f^{-1}(5x - 1) \]Now replace \(x\) in \(f^{-1}(x) = \frac{x+1}{5}\) with \(5x - 1\):\[ f^{-1}(f(x)) = \frac{(5x - 1) + 1}{5} \]Simplify:\[ = \frac{5x}{5} \]\[ = x \]Thus, \(f^{-1}(f(x)) = x\) is proven.

Step 4: Conclusion

Since both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold true, the functions \(f(x) = 5x - 1\) and \(f^{-1}(x) = \frac{x+1}{5}\) are indeed inverses of each other.

Key Concepts

Function Composition

The concept of function composition involves creating a new function by applying one function to the results of another. Suppose you have two functions, \( f(x) \) and \( g(x) \). The composition, \( (f \circ g)(x) \), is defined as \( f(g(x)) \). This means you first evaluate \( g(x) \) and then apply the function \( f \) to that result.

This concept is essential when dealing with inverse functions. Two functions, \( f(x) \) and \( f^{-1}(x) \), are considered inverses if their compositions yield the identity function. This is mathematically represented as \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

In simpler terms, if you apply one function to a number and then use its inverse, you should end up with the original number. Understanding this is key to verifying that two functions are inverses.

Algebra

Algebra plays a central role when working with functions and their inverses. It involves manipulating symbols and equations to establish equalities, which is a critical skill in this exercise.

For instance, in the verification of inverse functions, you often need to substitute variables and simplify expressions. This can involve operations like addition, subtraction, multiplication, and division.

Take the function \( f(x) = 5x - 1 \) and its presumed inverse \( f^{-1}(x) = \frac{x+1}{5} \). To confirm these are inverses, you must substitute and simplify expressions using algebraic methods. Such as replacing \( x \) in \( f(x) \) with \( f^{-1}(x) \) and simplifying to show \( f(f^{-1}(x)) = x \). Mastery of algebraic manipulation is essential for this process.

Intermediate Algebra

Intermediate algebra bridges the gap between basic algebra and more advanced topics like calculus. It includes an array of skills crucial for understanding functions, solving equations, and grasping inverse functions.

It’s important to familiarize yourself with solving linear equations, manipulating fractions, and handling inequalities. These are significant when verifying whether functions are inverses because they help us solve equations efficiently.

Consider how simplification in this context often involves canceling terms and isolating variables. For example, in our step-by-step solution, we simplified \( f(f^{-1}(x)) = 5 \left(\frac{x+1}{5}\right) - 1 \) directly to \( x \). This requires understanding how operations affect expressions and the ability to simplify fractions.

Intermediate algebra ensures that you can execute these steps smoothly and verify the properties of functions accurately.