Problem 17 - 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. $$(f+g)(1)$$

Step-by-Step Solution

Answer

The result of the function addition (f+g)(1) is 1.

Step 1: Understand Function Addition

Function addition involves adding the expressions of two functions. In this case, the functions are f(x) and g(x). Let's write out these functions again: \( f(x) = -x^{2} + x \) and \( g(x) = \frac{2}{x+1} \).

Step 2: Substitute the Value of X

Next, substitute x = 1 into both the functions. We get: \( f(1) = -1^{2} + 1 = 0 \) and \( g(1) = \frac{2}{1+1} = 1 \).

Step 3: Add the Function Values

Finally, add the respective values of f(1) and g(1) which are 0 and 1. Performing the addition, we get: \((f + g)(1) = f(1) + g(1) = 0 + 1 = 1.\)

Key Concepts

Composite Functions

When studying mathematics, one essential concept to grasp is that of composite functions. These are formed when one function is applied to the result of another function. The notation used is \( (f \circ g)(x) = f(g(x)) \) where \( g(x) \) is evaluated first, and then that result is used as the input for \( f(x) \).

Understanding the composition of functions is crucial for advanced concepts in calculus, such as chain rule in differentiation. It's essential to recognize that the order in which functions are composed matters; \( f \circ g \) is not the same as \( g \circ f \) in most cases. Hence, when dealing with composite functions, careful attention must be given to the order of operations.

Function Evaluation

The process of function evaluation is about finding the output of a function given a specific input. This seemingly simple process is at the heart of understanding how functions behave. For the function \( f(x) = -x^2 + x \) mentioned in the exercise, evaluating it at \( x = 1 \) means replacing every instance of \( x \) in the expression with 1, leading to \( f(1) = -1^2 + 1 = 0 \).

The skill of function evaluation is not just limited to plugging in a value; it includes understanding the behavior and properties of functions, such as their domain and range. Mastery of function evaluation will also assist students in analyzing more complex mathematical scenarios such as limit processes in calculus.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). A simple example is \( -x^2 + x \) from function \( f(x) \) in our exercise. These expressions can become complex and require an understanding of algebraic rules to simplify or manipulate.

To solve the given problem, knowledge of how to handle rational expressions like \( g(x) = \frac{2}{x+1} \) is also necessary. When we perform function addition, as in the exercise, we add two algebraic expressions together. In the case of functions \( f(x) \) and \( g(x) \) at \( x = 1 \) the correct algebraic process gives us \( (f + g)(1) = f(1) + g(1) = 0 + 1 = 1. \) Understanding algebraic expressions and their manipulation is the cornerstone of algebra and essential to progressing in mathematics.