Problem 18 - College Algebra

Question

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. $$ f(x)=-3 x+5 \text { and } g(x)=\frac{x-5}{-3} $$

Step-by-Step Solution

Answer

The functions are inverses because \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Understanding Function Composition

To verify that two functions are inverses, we must show that their compositions equal the identity function. This means we compute \(f(g(x))\) and \(g(f(x))\), and both should simplify to \(x\).

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

Substitute \(g(x) = \frac{x-5}{-3}\) into \(f(x) = -3x + 5\). \[ f(g(x)) = f\left(\frac{x-5}{-3}\right) = -3\left(\frac{x-5}{-3}\right) + 5 \]Simplifying this:\[ f(g(x)) = (-3) \cdot \frac{x-5}{-3} + 5 = (x-5) + 5 = x \].Thus, \(f(g(x)) = x\).

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

Substitute \(f(x) = -3x + 5\) into \(g(x) = \frac{x-5}{-3}\).\[ g(f(x)) = g(-3x + 5) = \frac{(-3x+5) - 5}{-3} \]Simplifying this:\[ g(f(x)) = \frac{-3x}{-3} = x \].Thus, \(g(f(x)) = x\).

Step 4: Conclusion

Since both compositions \(f(g(x)) = x\) and \(g(f(x)) = x\) hold true, functions \(f(x)\) and \(g(x)\) are indeed inverse functions.

Key Concepts

Understanding Function Composition

Function composition is a fundamental concept in mathematics that helps determine how functions interact with one another. It involves plugging one function into another. This can be thought of as a function eating another function! To check if two functions, say \( f(x) \) and \( g(x) \), are inverses, we use function composition. The key here is that when you compose the two functions, both \( f(g(x)) \) and \( g(f(x)) \) should return back to the original input \( x \).

• If \( f(g(x)) = x \), the result tells us that \( g(x) \) effectively "undoes" the work of \( f(x) \), returning us to the starting point.
• Similarly, if \( g(f(x)) = x \), it means \( f(x) \) "undoes" \( g(x)\) in the same way.

This symbiotic relationship is what defines inverse functions. Think of them as mathematical dance partners, perfectly coordinated in bringing each other back to the starting position. Use this composition process as a test every time you need to confirm the inverse relationship between two functions.

Identity Function and Its Role

The identity function, denoted usually by \( I(x) = x \), is like a pure reflection in math; it returns any input exactly as it is. When examining the inverse nature of two functions, function compositions should yield the identity function.
In simpler terms, if you feed the identity function with a number, it spits out the same number. It's the 'do-nothing' function but essential for determining inverses. If we compute both \( f(g(x)) \) and \( g(f(x)) \) using function composition and get \( x \) each time, this demonstrates the functions act as each other's inverse.

• The identity function's role here boosts our confidence that we've correctly computed the compositions.
• It assures us that the original functions get nullified to the initial input \( x \).

Comprehending the identity function's simplicity will highlight the elegance of inverse functions and give you a solid foundation for unraveling deeper math concepts.

Exploring Algebraic Functions

Algebraic functions, like those given in the exercise \( f(x) = -3x + 5 \) and \( g(x) = \frac{x-5}{-3} \), are made up of polynomials and rational expressions that follow set rules in their operations. When determining if they are inverses, algebraic manipulation is key.
To solve and confirm the inverse relationship, we substitute one function inside the other and simplify using standard algebra techniques. This involves operations like:

• Adding and subtracting constants.
• Multiplying and dividing terms.
• Simplifying fractions by canceling common terms.

By mastering these techniques, you can navigate through similar problems with confidence. Remember, underpinning all of this is the logical structure of algebra. It allows you to reliably work through function compositions, guiding you to the realization of when two functions are inverses.