Problem 19 - 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 (g+h)(0) is 3.

Step 1: Understand Function Addition

For two given functions, say g(x) and h(x), their addition (g+h)(x) is defined as g(x) + h(x). This involves simply taking the algebraic sum of the expressions defining the functions.

Step 2: Addition of Functions g and h

We apply the addition definition to functions g(x) and h(x). Therefore, (g+h)(x) = g(x) + h(x) = (2/(x+1)) + (-2x+1). Simplifying this expression gives us (g+h)(x) = 2/(x+1) - 2x + 1.

Step 3: Evaluate the Resultant Function at x = 0

Evaluate the resultant function from Step 2 at x = 0. (g+h)(0) = 2/(0+1) - 2*0 + 1. Simplifying this expression gives us the final answer of 3.

Key Concepts

Algebraic Sum of Functions

Understanding the algebraic sum of functions is critical for discussing how different pieces of mathematical information can be combined. In the context of functions, this process entails adding corresponding outputs from two or more functions when these functions have a common domain.

Imagine you have two functions, like pieces of a puzzle. Each function has its own set of rules that translates an input into an output. When we talk about the algebraic sum, we're essentially combining these rules. If your functions are defined by the expressions 'f(x)' and 'g(x)', their algebraic sum is simply denoted as '(f+g)(x)' and found by adding together the outputs: \[(f+g)(x) = f(x) + g(x).\]
This concept can extend to any number of functions. It's like a group project where each function brings its own contribution to the table—and the algebraic sum is the collaborative final product.

Evaluating Functions

Evaluating functions is akin to making a function come alive by giving it a number to transform. This can seem daunting, but it's just like following a cooking recipe. You have specific steps to create a dish; here, you have specific rules to get a result from an input number.

To evaluate a function, you substitute the input value, often called the 'x' value, into the function's equation. For instance, if our function is 'f(x) = 2x + 3', and we want to evaluate it at x = 5, we replace every 'x' in the function with 5, resulting in \[f(5) = 2(5) + 3 = 10 + 3 = 13.\]
Evaluating functions doesn't just tell us what they output for specific inputs; it also helps us understand the behavior of the function at those points—whether it's increasing, decreasing, or staying constant.

Function Operations

Beyond addition, function operations include subtraction, multiplication, and division. These are the tools that help us build complex mathematical structures from simpler ones.

For example, consider the functions 'f(x)' and 'g(x)' again. To multiply them, you would calculate \[(f \cdot g)(x) = f(x) \cdot g(x).\]
Just as you can multiply numbers, you multiply the outputs of the functions point by point. Similarly, to find the difference or quotient of two functions, you would subtract or divide their outputs for each input in their domain. Furthermore, function operations are subject not only to algebraic considerations but also to the conditions of the functions involved—like avoiding division by zero.

Importance of Function Operations

These operations are pivotal in calculus, where they lead to the concepts of derivatives and integrals. They're also fundamental in algebraic modeling of real-world scenarios, where various factors are often combined to understand a situation holistically. These concepts reinforce the interconnected nature of mathematics and its ability to reflect complex relationships through seemingly simple equations.