Question

Find the domain of \(h(x)=\frac{3 x}{x^{2}-9}\)

Short answer

The domain of \( h(x)=\frac{3 x}{x^{2}-9} \) is \( (-\infty,-3) \cup (-3,3) \cup (3,\infty) \).

Step-by-step solution

Identify the denominator

To find the domain of the function, start by identifying the denominator of the function. The denominator of the function is given by the expression under the fraction line. For the function \[ h(x)=\frac{3 x}{x^{2}-9} \] the denominator is \[ x^{2}-9. \]

Set the denominator equal to zero

To determine the values of \( x \) that make the function undefined, set the denominator equal to zero and solve for \( x \). \[ x^{2}-9=0 \]

Solve \( x^{2}-9=0 \)

Factor the equation to find the values that make the denominator zero. \[ x^{2}-9 = (x-3)(x+3) = 0 \] Solve each factor: \[ x-3=0 \implies x=3 \] \[ x+3=0 \implies x=-3 \]

Determine the domain

The domain of \( h(x) \) includes all real numbers except those that make the denominator zero, which we found to be \( x=3 \) and \( x=-3 \). Thus, the domain of \( h(x) \) is all real numbers except \( x=3 \) and \( x=-3 \). In interval notation, the domain is written as: \[ (-\infty,-3) \cup (-3,3) \cup (3,\infty) \]

Key concepts

denominators in algebra

In algebra, the denominator is the term located below the fraction line in a rational expression. It plays a crucial role because it determines the values for which the expression is defined. For example, in the function \[ h(x)=\frac{3 x}{x^{2}-9}, \]the denominator is \( x^{2}-9 \). The expression is undefined when the denominator is zero because division by zero is not possible. Therefore, identifying the denominator is an essential step in finding the domain of any rational function.

solving quadratic equations

Quadratic equations take the form \( ax^2 + bx + c = 0 \). These can often be solved through factoring, completing the square, or using the quadratic formula. For our specific problem, we had to solve the quadratic equation from the denominator, \( x^2 - 9 = 0 \). Factorization is a convenient method here:

• Recognize that \( x^2 - 9 \) is a difference of squares.
• Rewrite it as \( (x - 3)(x + 3) = 0 \).
• Set each factor to zero: \( x - 3 = 0 \) and \( x + 3 = 0 \).
• Solve each equation to get \( x = 3 \) and \( x = -3 \).

Since these values make the denominator zero, they are excluded from the domain.

interval notation

Interval notation is a way of writing subsets of the real number line. It describes the set of numbers between a starting and ending point. Here’s how it works for our domain:

• The denominator \( x^2 - 9 \) equals zero at \( x = 3 \) and \( x = -3 \). So these values are excluded.
• The domain includes all other real numbers.
• We express this in interval notation as \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \).

This notation is useful because it succinctly captures which ranges of the real numbers are valid for the function.