Problem 78 - Precalculus Mathematics for Calculus

Question

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$f(x)=\log x, \quad g(x)=x^{2}$$

Step-by-Step Solution

Answer

\( f \circ g = \log(x^2) \) with domain \( \mathbb{R} \setminus \{0\} \); \( g \circ f = (\log x)^2 \) with domain \( (0, \infty) \).

Step 1: Understand the Functions

Given two functions, where \( f(x) = \log x \) and \( g(x) = x^2 \), we need to find the compositions \( f \circ g \) and \( g \circ f \) and determine their domains.

Step 2: Find \( f \circ g \)

\( f \circ g \) means the function \( f(g(x)) \). We substitute \( g(x) = x^2 \) into \( f(x) \), so \( f(g(x)) = \log(x^2) \). Therefore, \( f \circ g = \log(x^2) \).

Step 3: Determine the Domain of \( f \circ g \)

For \( f \circ g = \log(x^2) \) to be defined, the input \( x^2 \) must be positive because the logarithm is defined for positive numbers only. Thus, \( x^2 > 0 \), which holds for all \( x eq 0 \). The domain of \( f \circ g \) is \( \mathbb{R} \setminus \{0\} \).

Step 4: Find \( g \circ f \)

\( g \circ f \) means the function \( g(f(x)) \). We substitute \( f(x) = \log x \) into \( g(x) \), resulting in \( g(f(x)) = (\log x)^2 \). Thus, \( g \circ f = (\log x)^2 \).

Step 5: Determine the Domain of \( g \circ f \)

Since \( g \circ f = (\log x)^2 \), and the logarithm \( \log x \) is defined only for positive \( x \), the domain of \( g \circ f \) is \( x > 0 \). Therefore, the domain is \( (0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all possible input values (or "x" values) that allow the function's formulas to work correctly without any mathematical errors. Understanding the domain is crucial because it tells us where the function is valid or applicable. When dealing with composite functions like in our original exercise, we blend two functions and need to consider the domains of both components:

• First, look at each function individually to understand its domain.
• Then, assess how combining them might restrict the domain further due to function composition rules.

In the provided exercise, we found two compositions: - For the composition \( f \circ g \), we needed \( x^2 \) to be positive, which occurs for every real number except zero. Thus, the domain of \( f \circ g \) is all real numbers except zero, written as \( \mathbb{R} \setminus \{0\} \).
- For \( g \circ f \), because \( \log x \) requires x to be more than zero, the domain is \( (0, \infty) \). Understanding these constraints helps prevent using undefined values, thereby avoiding errors in calculations.

Logarithmic Function

A logarithmic function is centered around the concept of the logarithm, which is essentially the inverse of exponentiation. The basic idea is solving for the power to which a base number, often 10 or \( e \) (Euler's number), must be raised to yield another number. Commonly seen as \( \log(x) \), this function helps us transition from exponential growth back to its roots.

• The key property of the logarithmic function is that it only exists for positive input values, due to its inverse relationship with exponentials.
• Logarithms are undefined for zero and negative numbers, as no exponentiation of these values can produce positive outcomes normally.

This principle is significant when dealing with compositions involving logarithms, as seen in \( f \circ g(x) = \log(x^2) \). Despite squaring x, ensuring non-negativity, logarithmic constraints still exclude zero, which affects the function's domain.
Mastery of logarithms empowers better understanding of real-world phenomena like scientific computations, indicators of scale, and growth measurements.

Quadratic Function

Quadratic functions are equations of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). They are graphically represented by parabolas, which can either open upwards or downwards depending on the sign and value of \( a \).

• Quadratics are special due to their distinctive parabolic curve, determined by their vertex, axis of symmetry, and opening direction.
• The domain of any quadratic function is all real numbers, \( \mathbb{R} \), because a quadratic expression can accept any real x-value.

In our exercise, \( g(x) = x^2 \), follows this pattern. Despite its free domain individually, when combined like in \( f \circ g \), new domain limitations may appear due to other function constraints like those of the logarithmic component.
Quadratics are fundamental in various areas from physics to finance, providing a foundation for modeling curves, optimizing processes, and more.