Problem 78 - Algebra and Trigonometry

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

\(f \circ g(x) = 2 \log x\), domain \(x \neq 0\); \(g \circ f(x) = (\log x)^2\), domain \(x > 0\).

Step 1: Understanding Composite Functions

To find the functions \(f \circ g\) and \(g \circ f\), we need to apply one function to the result of another. This means substituting the output of one function into the input of the other.

Step 1: Calculate \(f \circ g\)

The function \(f \circ g\) means \(f(g(x))\). Substitute \(g(x) = x^2\) into \(f(x) = \log x\), which gives us \(f(g(x)) = \log(x^2)\).

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

Using the property of logarithms, \(\log(x^2) = 2 \log x\). Therefore, \(f \circ g(x) = 2 \log x\).

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

For \(f \circ g(x) = 2 \log x\), the expression inside the logarithm, \(x^2\), must be greater than 0. This means \(x eq 0\). So, the domain of \(f \circ g\) is all real numbers except 0: \(x \in (-\infty, 0) \cup (0, \infty)\).

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

The function \(g \circ f\) means \(g(f(x))\). Substitute \(f(x) = \log x\) into \(g(x) = x^2\), which gives us \(g(f(x)) = (\log x)^2\).

Step 5: Simplify \(g \circ f\)

The expression \((\log x)^2\) is already simplified. Therefore, \(g \circ f(x) = (\log x)^2\).

Step 6: Determine the Domain of \(g \circ f\)

For \(g \circ f(x) = (\log x)^2\), the input to \(f\), \(x\), must be greater than 0 so that \(\log x\) is defined. Therefore, the domain of \(g \circ f\) is all positive real numbers: \(x \in (0, \infty)\).

Key Concepts

Understanding Function Composition

Composite functions are an essential part of algebra and calculus, and they involve the combination of two functions where one function is applied to the result of another. To form a composite function, you take two functions, say \( f(x) \) and \( g(x) \). The notation \( f \circ g \) or \( g \circ f \) represents these composite functions. In this notation, \( f \circ g(x) = f(g(x)) \), which means you first apply \( g(x) \), then take its output and use it as an input to \( f(x) \). Similarly, \( g \circ f(x) = g(f(x)) \), which means you apply \( f(x) \) first and use its output as the input for \( g(x) \).

• For \( f \circ g(x) = f(g(x)) \), substitute \( g(x) \) into \( f(x) \).
• For \( g \circ f(x) = g(f(x)) \), substitute \( f(x) \) into \( g(x) \).

Understanding this concept allows you to manipulate complex functions and find the relationships between them effectively.

Defining the Domain of a Function

The domain of a function is the set of all possible inputs for which the function is defined. When dealing with composite functions, finding the domain can be a little more involved.
For example, if you have \( f(x) = \log x \), the domain of \( f \) is all positive numbers \( x > 0 \) because the logarithm is only defined for positive numbers. Similarly, the function \( g(x) = x^2 \) has the entire set of real numbers as its domain, \( x \in \mathbb{R} \), because you can square any real number.
When composing functions, the domain of the composite function \( f \circ g(x) \) or \( g \circ f(x) \) will depend on the domains of the original functions. Particularly:

• For \( f \circ g(x) = 2 \log x \), the expression inside the logarithm, \( x^2 \), must be positive, excluding zero \( x eq 0 \), so the domain is all real numbers except zero.
• For \( g \circ f(x) = (\log x)^2 \), the input must also be positive to define \( \log x \), so the domain is all positive numbers \( x > 0 \).

Every composite function's domain must respect the constraints from both \( f(x) \) and \( g(x) \).

Exploring Logarithmic Functions

Logarithmic functions are a type of mathematical function where the variable is inside a logarithm. The natural logarithm function, denoted usually by \( \log x \) when the base is natural (commonly base \( e \)), is used extensively in both pure and applied mathematics. The primary characteristic is that they are the inverse of exponential functions. If \( y = e^x \), then \( x = \log y \).
Key properties of logarithmic functions include:

• The domain of \( \log x \) is \( x > 0 \), since only positive numbers have real logarithms.
• Logarithms have the special property \( \log(a^b) = b \log a \), which can be used in simplifying expressions like \( \log(x^2) = 2 \log x \).

In composite functions, logarithmic functions often influence the domain because their inputs cannot be zero or negative. Understanding how to manipulate expressions and solve logarithmic equations is vital. Recognizing when and how to apply logarithms can help simplify complex mathematical problems and express functions in alternative forms.