Problem 28 - Calculus: One and Several Variables

Question

Form the composition \(f \circ g\) and give the domain. $$f(x)=1 /(x-1), \quad g(x)=x^{2}$$

Step-by-Step Solution

Answer

\(f \circ g(x) = 1/(x^{2} - 1)\) and the domain is all real numbers except -1 and 1.

Step 1: Find the composition

The composition \(f \circ g\) is obtained by substituting \(g(x)\) into \(f(x)\). When we do this, we obtain \(f \circ g(x) = f(g(x)) = f(x^{2}) = 1/(x^{2} - 1)\)

Step 2: Find the domain

The function, \(1/(x^{2} - 1)\), is undefined when \(x^{2} - 1 = 0\). Solving the equation \(x^{2} - 1 = 0\) yields \(x = -1, 1\). Therefore, the domain of the function \(f \circ g\) is all real numbers except -1 and 1.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values that allow the function to work without any restriction. It is essentially the set of all x-values for which the function is defined and provides a real-number output.

For example, when dealing with rational functions such as \( f(x) = \frac{1}{x-1} \), the domain excludes values that make the denominator zero. In this case, the value x = 1 is not part of the domain because it would create division by zero, which is undefined. Calculating the domain of composite functions like \( f \circ g \) requires substituting one function into the other and then determining the domain of the resulting function.

In the given exercise, after forming the composite function, we must exclude the x-values that make the denominator of the composite function \( f(g(x)) \) equal to zero, in this case, x-values -1 and 1. Thus, the domain of the composite function \( f \circ g \) is all real numbers except for -1 and 1.

Composite Functions

Composite functions involve two functions being combined in such a way that the output of one function becomes the input of another. This is often denoted as \( f \circ g \) and read as 'f composed with g'.

To form a composite function, you substitute the entire second function, \( g(x) \), into the first function, \( f(x) \), wherever there is an x. In mathematical terms, \( (f \circ g)(x) = f(g(x)) \). This creates a new function that demonstrates a specific relationship between x and the final output.

Composite functions are useful for modeling complex scenarios where one situation directly affects another. Calculating the composite function requires careful substitution, as shown in the provided exercise. The correct substitution and simplification yield \( f \circ g(x) = \frac{1}{{x^2} - 1} \), which combines the characteristics of both original functions.

Rational Functions

Rational functions are fractions that involve polynomials in both numerator and denominator. The general form of a rational function is \( r(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).

One of the key characteristics of rational functions is that they may not be defined for all real numbers, due to potential zeros in the denominator. To determine the points where the function is undefined, you set the denominator equal to zero and solve for x, resulting in the exclusion of these x-values from the domain.

In our exercise, \( f(x) \) is a rational function where the denominator \( x - 1 \) cannot be zero, so we exclude 1 from the domain of \( f \). However, when we compose it with \( g(x) = x^2 \), the resulting domain excludes both -1 and 1, showing how composition affects the domain.