Problem 67 - Precalculus : Building Concepts and Connections

Question

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=-x^{2}+1 ; g(x)=x+1$$

Step-by-Step Solution

Answer

The composition \( (f \circ g)(x) = -x^2 - 2x \) and \( (g \circ f)(x) = -x^2 + 2 \). The domains of both compositions are the set of all real numbers.

Step 1: Compute \( (f \circ g)(x) \)

To find \( (f \circ g)(x) \), insert function \( g(x) \) into function \( f(x) \). The function \( g(x) = x+1 \), insert this into \( f(x) = -x^2 + 1 \) to obtain \( f(g(x)) = -(x+1)^2 + 1 \). Simplifying this expression, you will get \( f(g(x)) = -x^2 - 2x \).

Step 2: Compute \( (g \circ f)(x) \)

To find \( (g \circ f)(x) \), insert function \( f(x) \) into function \( g(x) \). The function \( f(x) = -x^2 + 1 \), insert this into \( g(x) = x+1 \) to obtain \( g(f(x)) = -x^2 + 1 + 1 = -x^2 + 2 \).

Step 3: Finding the domains of \( f \circ g \) and \( g \circ f \)

The domain of a function is the set of all real numbers where the function is defined. The domain of \( (f \circ g)(x) = -x^2 - 2x \) is the set of all real numbers, because the expression is defined for all \( x \). Likewise, the domain of \( (g \circ f)(x) = -x^2 + 2 \) is also the set of all real numbers.

Key Concepts

Understanding Function Domains

The domain of a function refers to all possible input values for which the function is defined. For composite functions such as \( (f \circ g)(x) \) or \( (g \circ f)(x) \), understanding the domain involves considering the domains of the individual functions \( f(x) \) and \( g(x) \), along with the composition itself.

In our exercise, we compute:\( (f \circ g)(x) \) to be \(-x^2 - 2x\). This is a quadratic expression with no undefined values when considering real numbers. Hence, the domain is all real numbers. Always remember:

• Polynomials like \( -x^2 - 2x \) are continuous on the entire real line.
• Look out for special situations such as division by zero or square roots of negative numbers which can restrict domains.

Similarly, \( (g \circ f)(x) \) gives \(-x^2 + 2\). Again, this is a quadratic expression, also defined for all real numbers. Thus, its domain is all real numbers too. When identifying domains, it's crucial to ensure that the function is real for all inputs in the domain considered, which in these examples, it is.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basic building blocks of mathematical equations and functions. In the context of our problem, we are dealing with \( f(x) = -x^2 + 1 \) and \( g(x) = x+1 \), which are both polynomial expressions.

Polynomials are a simple type of algebraic expression that consists of variables raised to whole number powers and their coefficients. When combining functions to form composite functions such as \( (f \circ g)(x) \), we end up with a new algebraic expression. Here's what happens:

• Insert \( g(x) \) into \( f(x) \): First substitute \( x+1 \) wherever \( x \) appears in \( f(x) \), resulting in \(-(x+1)^2 + 1\).
• Simplify: Carefully expand and simplify the expression to \(-x^2 - 2x\). Handling negative signs and powers correctly is key here.

This approach of substituting and simplifying is fundamental in algebra. Understanding how to manipulate and simplify such expressions is central to mastering algebra and functions.

Working with Real Numbers

Real numbers are the set of all numbers that can be found on the number line, including both rational (like \( 1, \frac{1}{2}, \frac{3}{4}\)) and irrational numbers (like \( \sqrt{2}, \pi \)). These form the foundational elements for most mathematical concepts, especially in algebra and calculus.

In the context of this exercise, both the domains of \( f \circ g \) and \( g \circ f \) consist of all real numbers. This is because their expressions, \(-x^2 - 2x\) and \(-x^2 + 2\), are continuous and defined for every real input value. Real numbers allow us to understand how functions behave in a continuous manner and are crucial when discussing domains:

• Continuous: Every value within a given interval on the real number line is valid.
• Unified set: Combines both rational and irrational numbers, making it a complete number system for practical purposes.

This ensures that the discussion of domains is both inclusive and complete when only considering real numbers. No unexpected breaks, gaps, or asymptotes hinder the calculation or graphing of these functions over real numbers.