Problem 25 - 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. $$(g-h)(-2)$$

Step-by-Step Solution

Answer

-7

Step 1: Calculate Function g(-2)

First, substitute -2 into the function \(g(x)=\frac{2}{x+1}\). This gives \(g(-2)=\frac{2}{-2+1}=-2\).

Step 2: Calculate Function h(-2)

Substitute -2 into the function \(h(x)=-2x +1\). This gives \(h(-2)=-2*(-2)+1=5\).

Step 3: Evaluate (g-h)(-2)

Now, apply the function operation (g-h)(-2).This gives us \(-2-5=-7.\)

Key Concepts

Function Evaluation

Function evaluation involves finding the output of a function when a specific input is provided. To find this, substitute the input value into the function's formula. For instance, given a function \( f(x) = x^2 + 3x + 2 \), to evaluate \( f(3) \), plug in 3 for every instance of \( x \):
\[ f(3) = 3^2 + 3(3) + 2 = 9 + 9 + 2 = 20\]This process allows you to compute exactly what the value of the function is for any given input. For the functions in the exercise:

• We substituted \(-2\) into the formula for \( g(x) \) to find \( g(-2) \).
• We also did the same for \( h(x) \) to find \( h(-2) \).

By calculating these, we determine the values of \( g \) and \( h \) at a specific point.

Algebraic Functions

Algebraic functions are made up of sums, products, differences, and roots of variables. These functions can be expressed in terms of basic arithmetic operations.
For example, in the function \( f(x) = -x^2 + x \), it involves:

• A power operation with the \( x^2 \) term.
• A multiplication by -1, represented by \( -x^2 \).
• A linear addition, represented by \( +x \).

Algebraic functions can be very intricate or quite straightforward. They are an essential part of high school algebra and beyond, helping us model real-world situations. In the given problem, the functions \( g(x) = \frac{2}{x+1} \) and \( h(x) = -2x + 1 \) are both algebraic functions that utilize division and multiplication respectively. Understanding these operations is crucial for performing correct evaluations and manipulations of functions.

Composite Functions

Composite functions are created when one function is applied to another function's output. This process involves stacking functions to create new operations.
Consider two functions \( f(x) \) and \( g(x) \). If you want to evaluate the composite function \( f(g(x)) \), you take the result from \( g(x) \) and input it into \( f(x) \). For instance, if \( f(x) = x + 1 \) and \( g(x) = x^2 \), then:

• First, evaluate \( g(x) \), let's say \( g(2) = 4 \).
• Then evaluate \( f(g(2)) = f(4) = 4 + 1 = 5 \).

In the context of the exercise, the operation \( (g-h)(-2) \) is not a typical composite function but an application of operation functions where you apply subtraction after evaluating both functions separately. The important skill is understanding how operations on functions are applied to the inputs, evidenced by subtracting the evaluated values of \( g(-2) \) from \( h(-2) \). This solidifies comprehension of how to handle combinations of different function types.