Problem 32 - Precalculus : Building Concepts and Connections

Question

In Exercises \(17-40,\) let \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1},\) and \(h(x)=-2 x+1 .\) Evaluate each of the following. $$(g h)(0)$$

Step-by-Step Solution

Answer

The value of the function composition \( (g \circ h)(0) \) is 1.

Step 1: Compute the Value of Inner Function at the Point

First, we need to compute the value of the inner function -- which is \( h \) -- at the given input, which is 0. So we compute \( h(0) \). Substituting \( x = 0 \) into \( h(x) \), we get: \( h(0) = -2*0 + 1 = 1. \)

Step 2: Substitute the Result into the Outer Function

Next, we take the result from Step 1 and use it as the input into the outer function \( g \). We compute \( g(h(0)) \), which is equivalent to \( g(1) \). Substituting \( x = 1 \) into \( g(x) \), we get: \( g(1) = 2/(1 + 1) = 1. \)

Key Concepts

Function Evaluation

Function evaluation means finding the value of a function at a specific point. In the context of composite functions, this evaluation is multi-step as it typically involves more than one function. Here, the composite function utilizes multiple functions such as \( f(x) \), \( g(x) \), and \( h(x) \). To evaluate a composite function, you begin by finding the value of the innermost function first. The reason being, the result directly affects the next function in line.

For example, if you're given a composite function \((g \, h)(x)\), you would first evaluate \( h(x) \). Once you have \( h(x) \), you use that result as the input for \( g(x) \). This step-by-step procedure ensures that you systematically find the value of the overall composite function.

Inner and Outer Functions

In composite functions, you have both inner and outer functions, which determine the order of evaluation. The inner function is where you start your evaluation. It is the first function that you will plug the initial variable into.

The outer function comes into play once the inner function has been evaluated. It takes the output of the inner function as its input. This layering approach, where functions are stacked within each other, is the hallmark of composite functions such as \((g \, h)(x)\).

• Inner Function: \( h(x) = -2x + 1 \)
• Outer Function: \( g(x) = \frac{2}{x+1} \)

Understanding the distinction between them is crucial because the output from the inner function becomes the input for the outer function.

Step-by-Step Problem Solving

When you solve problems involving composite functions, a step-by-step approach is crucial. This approach ensures that you don't miss any steps, leading to a correct result. Typically, the process can be broken down into organized steps, as follows:

1. **Evaluate the Inner Function:** Start by assessing the inner function at the given point, like calculating \( h(0) \) from the example, which yields 1.
2. **Plug Value into Outer Function:** Next, use this result as the input for the outer function, so you compute \( g(1) \), resulting in 1.

By following these steps individually and in order, you simplify what could otherwise be a complex task. With practice, this structured method of solving composite functions not only becomes second nature but also deeply ingrains your understanding of each component's role in the larger solution.