Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

The function in Example 9 can be decomposed in other ways. For which of the following pairs of functions is \(h(x)=\frac{1}{(x-2)^{2}}\) equal to \(f(g(x)) ?\) (a) \(g(x)=\frac{1}{x-2}\) and \(f(x)=x^{2}\) (b) \(g(x)=x^{2}\) and \(f(x)=\frac{1}{x-2}\) (c) \(g(x)=(x-2)^{2}\) and \(f(x)=\frac{1}{x}\)

Step-by-Step Solution

Answer

Option (a) is the correct choice.

Step 1: Evaluate Option (a)

For option (a), we have \(g(x)=\frac{1}{x-2}\) and \(f(x)=x^{2}\). Evaluate \(f(g(x))\) by replacing \(g(x)\) into \(f(x)\). Thus, \(f(g(x))=\left(\frac{1}{x-2}\right)^{2}=\frac{1}{(x-2)^{2}}\). So for option (a), \(f(g(x))=h(x)\). Hence, option (a) is correct.

Step 2: Evaluate Option (b)

For option (b), we have \(g(x)=x^{2}\) and \(f(x)=\frac{1}{x-2}\). Evaluate \(f(g(x))\) by replacing \(g(x)\) into \(f(x)\). Thus, \(f(g(x))=\frac{1}{(x^{2}-2)}\), which is not equal to \(h(x)\). Hence, option (b) is incorrect.

Step 3: Evaluate Option (c)

For option (c), we have \(g(x)=(x-2)^{2}\) and \(f(x)=\frac{1}{x}\). Evaluate \(f(g(x))\) by replacing \(g(x)\) into \(f(x)\). Thus, \(f(g(x))=\frac{1}{(x-2)^{2}}\), which is not equal to \(h(x)\). Hence, option (c) is incorrect.

Key Concepts

Understanding Composite Functions

Composite functions are like putting one function inside another. Imagine a box within a box—one action depends on another step. For example, if you have functions \(f\) and \(g\), the composite function is written as \(f(g(x))\).

• How it works: First, solve for \(g(x)\). Imagine \(g(x)\) as a middle process.
• Next, use the result of \(g(x)\) in \(f(x)\). This is like saying what happens after you open the middle box.

The purpose of a composite function is to create new relationships and operations by chaining existing ones. In our exercise, \(f(g(x))\) needed to result in \(h(x) = \frac{1}{(x-2)^{2}}\). This required finding the right function pair for \(f\) and \(g\). By understanding this layered process, you build a toolset for more complicated math problems.

Mastering Function Notation

Function notation acts like a label for operations and relationships. It simplifies expressions using a compact format. When you see something like \(f(x)\), you're seeing function notation in action.

• What it means: The expression tells you which input goes into the function and what name or label to use.
• Instead of writing recurrent operations each time, function notation ensures clarity.
• It's like having a code or shorthand for long operations.

In our exercise, the notation \(g(x)\), \(f(x)\), and \(h(x)\) indicated each function's role. Thinking of functions as machines that transform inputs helps. You put \(x\) in, and out comes \(g(x)\), \(f(x)\), or whatever result correspondingly. Each function handles its task as described in its compact notation.

The Art of Evaluating Functions

Evaluating functions is like following detailed instructions or a recipe. To evaluate means to compute the result of a function expression. For any functions \(f\) and \(g\), evaluating usually involves substitution.

• Start with the inside: Often, you begin with an innermost function. In composite scenarios like \(f(g(x))\), start by solving \(g(x)\).
• Use the result: Take the result from the first step and substitute it into the next function.
• Continue until complete: Repeat until you have satisfied all function layers.

In the original exercise, evaluating step-by-step showed that only option (a) matched \(h(x)\). Practice makes perfect when mastering this art, and breaking it into steps helps avoid errors. Function evaluation transforms abstract concepts into tangible solutions you can see and solve.