Question

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\frac{5}{x^{2}-4}, \quad g(x)=x+3\)

Short answer

The domain of \(f(x)\) is \(x \in R, x \neq 2, -2\), the domain of \(g(x)\) is all real numbers or \(x \in R,\) and the domain of \(f \circ g\) is \(x \in R, x \neq -1, -5\).

Step-by-step solution

Determine the Domain of Function \(f\)

The function \(f(x)\) is defined for all real values of \(x\) except the ones that make the denominator zero. Therefore, solve the equation \(x^2 - 4 = 0\) to find which values of \(x\) are not included in the domain. Solving the equation gives \(x = 2\) and \(x = -2\). Therefore, the domain of \(f(x)\) is all real numbers except 2 and -2 that is \(x \in R, x \neq 2, -2\).

Determine the Domain of Function \(g\)

The function \(g(x)\) is a linear function, and all linear functions are defined for all real values of \(x\). Therefore, the domain of \(g(x)\) is all real numbers, or \(x \in R\).

Determine the Domain of the Composite Function \(f \circ g\)

For the composite function \(f(g(x))\), first note that the output of \(g(x)\) is \(x + 3\), and this output should be in the domain of the function \(f(x)\). However, we previously found that \(f(x)\) is undefined for \(x = 2\) and \(x = -2\). Therefore, we set \(x + 3 = 2\) and \(x + 3 = -2\) and solve for \(x\) to find the values that are not included in the domain of the composite function. Solving these gives \(x = -1\) and \(x = -5\). Therefore, the domain of \(f(g(x))\) is all real numbers except -1 and -5, that is \(x \in R, x \neq -1, -5\).

Key concepts

Composite Functions

Composite functions involve combining two or more functions to create a new function. The process is similar to putting two machines in a row; the output from the first machine becomes the input for the second. In the given exercise, we deal with a composite function, written as \( f \circ g \), which means \( f(g(x)) \)—that is, we first apply \( g \) to the input \( x \) and then apply \( f \) to the result of \( g(x) \).

To find the domain of a composite function, we must ensure that every element in the domain of \( g \) maps to an element in the domain of \( f \) when \( g \) is applied. Thus, we must exclude any values from \( g \) that would lead to 'forbidden' values for \( f \). In simpler terms, avoid values for \( x \) that make the function undefined or lead to division by zero in the case of a rational function like \( f \).

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. The domain of a rational function such as \( f(x) = \frac{5}{x^{2}-4} \) consists of all real numbers except those that make the denominator zero.

The key step in determining the domain of a rational function is to set the denominator equal to zero and solve for \( x \)—these solutions are the values that must be excluded from the domain. For \( f(x) \), solving \( x^{2} - 4 = 0 \) gives \( x = 2 \) and \( x = -2 \) as the 'troublesome' values, hence they are not included in the domain of \( f \).

Linear Functions

Linear functions are the simplest type of function you will come across and have a constant rate of change, or slope. The general form of a linear function is \( g(x) = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

These functions graph as straight lines and are defined for all real numbers, meaning their domain is always the set of all real numbers, or \( x \in R \). For example, in the exercise, the function \( g(x) = x + 3 \) is linear and thus has no restrictions on its domain. All real number inputs are valid.

Real Numbers

Real numbers include all the numbers on the number line, comprising both rational numbers (like fractions and whole numbers) and irrational numbers (like \( \sqrt{2} \) and \( \pi \)). They can be positive, negative, or zero.

The concept of real numbers is crucial when discussing the domain of a function, as most functions in basic algebra have a domain that is a subset of the real numbers. For instance, when we exclude certain numbers from the domain of \( f(x) \) or \( g(x) \), we still talk within the realm of real numbers—identifying which real numbers are not permissible for a given function.