Problem 73 - Algebra 2

Question

Determine whether \(f(x)=x-2\) and \(g(x)=2 x\) are inverse functions.

Step-by-Step Solution

Answer

The functions \(f(x)=x-2\) and \(g(x)=2x\) are not inverse functions.

Step 1: Define Inverse Function

Two functions \( f(x) \) and \( g(x) \) are inverses of each other if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in their domain.

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

To find \( f(g(x)) \), substitute \( g(x) = 2x \) into \( f(x) = x - 2 \). This gives:\[f(g(x)) = f(2x) = 2x - 2\]For \( f \) and \( g \) to be inverses, this must simplify to \( x \). However:\[ 2x - 2 eq x\]so \( f(g(x)) \) does not equal \( x \).

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

Next, compute \( g(f(x)) \) by substituting \( f(x) = x - 2 \) into \( g(x) = 2x \). This gives:\[g(f(x)) = g(x-2) = 2(x-2) = 2x - 4\]For \( f \) and \( g \) to be inverses, this must simplify to \( x \). However:\[2x - 4 eq x\]so \( g(f(x)) \) does not equal \( x \).

Step 4: Conclusion

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \) holds true, \( f(x) \) and \( g(x) \) are not inverse functions.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics that involves the application of one function to the results of another. This is crucial to understanding inverse functions. When you compose two functions, you combine them such that the output of one function becomes the input of the other. For example, if you have two functions, say, \( f(x) \) and \( g(x) \), the composition of \( f \) with \( g \) is written as \( f(g(x)) \).

This concept can be visualized as linking two machines together. You feed an input into \( g(x) \), and whatever comes out becomes the input for \( f(x) \). The result is \( f(g(x)) \). In the exercise you have, you computed \( f(g(x)) \) as:

• Start with \( g(x) = 2x \).
• Plug \( 2x \) into \( f(x) = x - 2 \).

This gives \( f(g(x)) = 2x - 2.\) For two functions to be inverses, when you perform such composition, you should end up with \( x \), essentially reversing the effect of the first function with the second.

Domain of Functions

The domain of a function is the complete set of possible values of the independent variable, usually \( x \). In simpler terms, it’s all the x-values that can go into the function without causing any issues like division by zero or taking a square root of a negative number. Understanding the domain is crucial when dealing with inverse functions since for two functions to be inverses, their domains and co-domains (outputs) must match appropriately.

In the context of the exercise, knowing the domain of \( f(x) = x-2 \) and \( g(x) = 2x \) aids in seeing whether their compositions can produce the same x-values:

• For \( f(x) = x-2 \), the domain is all real numbers; you can subtract 2 from any number.
• Similarly, \( g(x) = 2x \) also accepts all real numbers as you can multiply any number by 2.

In this case, the functions' domains overlap sufficiently, allowing us to apply them in composition. However, the results of the composed functions \( f(g(x)) \) and \( g(f(x)) \) did not match exactly with \( x \), indicating these functions are not inverses.

Verifying Inverse Functions

To verify whether two functions are inverses, there are a few easy-to-apply steps that you should follow. Essentially, two functions \( f(x) \) and \( g(x) \) are inverses if both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for all \( x \) in their respective domains.

Let's break this down based on your exercise:

• Compute \( f(g(x)) \) and check if it simplifies to \( x \).
• Similarly, compute \( g(f(x)) \) and see if it also gives \( x \).
• If either composition does not simplify to \( x \), then \( f(x) \) and \( g(x) \) are not inverses.

In your example, \( f(g(x)) = 2x - 2 \) and \( g(f(x)) = 2x - 4 \). Both outputs differ from \( x \), so they don't meet the criteria for inverse functions.

Verifying inverse functions ensures that each function truly "undoes" the other. Meaning, applying one after the other returns you to your starting value.