Problem 14 - Precalculus : Building Concepts and Connections

Question

Verify that the given functions are inverses of each other. $$f(x)=\frac{1}{2} x+1 ; g(x)=2 x-2$$

Step-by-Step Solution

Answer

Yes, the functions f(x) and g(x) are inverses of each other.

Step 1: Compose the Functions

First, let's substitute g(x) into f(x), thus obtaining f(g(x)). g(x) is given as 2x - 2, replacing x in f(x) with g(x) we get \(f(g(x)) = 0.5*(2x - 2) + 1\)

Step 2: Simplify the equation for f(g(x))

Expanding and simplifying, we will get \(f(g(x)) = x - 1 + 1\), which simplifies further to \(f(g(x)) = x\). This proves half the requirement that f(g(x)) should be equal to x.

Step 3: Compose the functions in reverse

Now substitute f(x) into g(x) to find g(f(x)). f(x) is given as 0.5x + 1, so replacing x in g(x) with f(x) we get \(g(f(x)) = 2*(0.5x + 1) - 2\)

Step 4: Simplify the equation for g(f(x))

Expanding and simplifying, we will be left with \(g(f(x)) = x + 2 - 2\), which simplifies further to \(g(f(x)) = x\). This proves the second requirement that g(f(x)) should equal to x.

Step 5: Verification

Both f(g(x)) and g(f(x)) have been shown to equal x. Therefore, we can conclude that the given functions f(x) and g(x) are indeed inverses of each other.

Key Concepts

Verifying Inverse Functions

To understand if two given functions are inverses of each other, we must verify that the composition of one function with the other returns the original input variable. This means if you have two functions, say, function f and its alleged inverse g, then both compositions f(g(x)) and g(f(x)) must equal x for all values in the domain of these functions.

Using the functions from the original exercise, \( f(x) = \frac{1}{2}x + 1 \) and \( g(x) = 2x - 2 \) as examples, let's walk through the verification step by step. When we substitute function g into function f, \( f(g(x)) \) is supposed to simplify to \( x \). Likewise, substituting function f into function g, \( g(f(x)) \) should also simplify to \( x \).

Once this criterion is satisfied, as it was in the provided solution, we can confidently state that the functions f(x) and g(x) are inverses of each other.

Function Composition

Function composition involves creating a new function by applying one function to the result of another. The notation for the composition of two functions, f and g, is \( f(g(x)) \) or g(f(x)). It's like giving an input to function g and then using the output of g as the input to function f.

In our exercise, composing \( f \) with \( g \) means to evaluate f at the place where x is by plugging in the value of g(x), resulting in \( f(g(x)) = 0.5 * (2x - 2) + 1 \). Upon simplification, the equation simplifies to \( x \), which is exactly what we expect for an inverse relationship.

Composing functions provides insight into how one function modifies the effect of another and is a key concept in understanding complex real-world processes where multiple operations occur sequentially.

Simplifying Equations

Simplifying equations in precalculus often involves steps to reduce expressions to their simplest form through operations like combining like terms, distributing multiplication over addition, and canceling out terms where possible.

When simplifying \( f(g(x)) \) in the original exercise, we expanded \( 0.5*(2x - 2) + 1 \) to \( x - 1 + 1 \) and then combined like terms to arrive at the simplified form \( x \). Similarly, simplifying \( g(f(x)) \) required distributing the multiplication in \( 2*(0.5x + 1) - 2 \) and then cancelling out terms to reduce it down to \( x \).

Effective simplification requires a methodical approach to ensure that all possible reductions are made without altering the equation's meaning, leading to a more understandable form and making it easier to verify the properties of the functions involved, such as their inverses.