Problem 29 - 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. $$(f g)(3)$$

Step-by-Step Solution

Answer

The value of (f g)(3) is 0.25.

Step 1: Compute the function composition

The composition of two functions f and g, denoted as (f g)(x), is defined as f(g(x)). Therefore, (f g)(3) would be evaluated as such: \(f(g(3))\).

Step 2: Evaluate g at x=3

Substitute x=3 into the function g(x) to evaluate g(3). \n\[g(3) = \frac{2}{3+1} = \frac{2}{4} = 0.5 \]

Step 3: Substitute g(3) into f(x)

Substitute g(3) or 0.5 into the function f(x) to find f(g(3)). \n\[f(g(3)) = f(0.5) = -(0.5)^{2} + 0.5 = -0.25 + 0.5 = 0.25 \]

Key Concepts

Composition of Functions

In mathematics, the composition of functions refers to the process of applying one function to the results of another. This is usually denoted as \((f \circ g)(x)\), or simply \(f(g(x))\). To understand this, consider each function as a machine that takes an input and gives an output. The composition of functions is like connecting these machines where the output of the second one becomes the input to the first.

• Inside-Out Evaluation: Always evaluate from the innermost function outward. So, when given \(f(g(x))\), start by calculating \(g(x)\).
• Order Matters: The functions are not interchangeable. \(f(g(x))\) might not be equal to \(g(f(x))\).

Function composition is a fundamental concept in algebra and calculus, often used to build more complex functions or solve problems that require evaluating multiple functions in sequence. This operation allows for deeper analysis and understanding of function behaviors and relationships.

Evaluate Functions

Evaluating a function involves finding the output given a specific input. When we evaluate functions, we replace the variable, typically \(x\), with the given value and perform the necessary calculations. Let's break this process down:

• Substitution: Start by substituting the given value into the function wherever the variable appears.
• Perform Calculations: Follow the order of operations – parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

For example, to evaluate \(g(3)\), you substitute 3 into the equation for \(g(x)\). After substituting, you compute: \[g(3) = \frac{2}{3 + 1} = \frac{2}{4} = 0.5\] This methodical process helps in obtaining an accurate output from a function for a particular input.

Precalculus Exercise

Exercises like evaluating \((f g)(3)\) are common in precalculus, a mathematical course that sets the foundation for calculus. Such problems teach students how to work with different types of functions and understand their interrelations.

• Skill Development: These exercises enhance algebraic manipulation skills, logical reasoning, and a deeper understanding of how functions work together.
• Advanced Preparation: Mastery of these concepts is crucial for succeeding in calculus, where similar concepts are applied in more complex scenarios.

When faced with a problem like \((f g)(3)\), it's crucial to follow each step systematically: compute \(g(3)\) first, then find \(f(g(3))\). This approach helps reinforce the step-by-step procedures necessary for solving more complex mathematical problems in the future.