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Chapter 12: Problem 61

How do you find the domain of a rational function?

Short Answer

Expert verified
To find the domain of a rational function \(f(x) = \frac{P(x)}{Q(x)}\), first identify the denominator, \(Q(x)\). Then set \(Q(x) = 0\) to find the x-values that make the function undefined. Exclude these x-values from the domain, and express the domain in interval notation. For example, the domain of \(f(x) = \frac{2x + 1}{x^2 - 9}\) is \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

Step by step solution

01

Identify the function's denominator

First, identify the denominator of the rational function, which is the expression in the denominator of the fraction. Let the rational function be in the form \(f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials. In this case, the denominator is \(Q(x)\).
02

Set the denominator equal to zero

Next, set the denominator equal to zero by writing the equation \(Q(x)=0\). This equation represents the x-values that would make the function undefined.
03

Solve for x-values

Solve the equation \(Q(x) = 0\) for x. The solutions to this equation will be the values of x that we need to exclude from the function's domain since they would make the denominator equal to zero.
04

Express the domain in interval notation

Now that you have found the values of x to exclude, express the domain of the function in interval notation. The domain will include all values of x except for the solutions found in step 3. Example: Let鈥檚 find the domain of the rational function \(f(x) = \frac{2x + 1}{x^2 - 9}\).
05

Example Step 1: Identify the function's denominator

The denominator of the function is \(x^2 - 9\).
06

Example Step 2: Set the denominator equal to zero

Set the denominator equal to zero: \(x^2 - 9 = 0\).
07

Example Step 3: Solve for x-values

Solve the equation for x: \(x^2 - 9 = 0\) \((x - 3)(x + 3) = 0\) So, \(x = 3\) and \(x = -3\).
08

Example Step 4: Express the domain in interval notation

The domain of the function includes all x-values except 3 and -3. In interval notation, this can be written as: \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\). So, the domain of the rational function \(f(x) = \frac{2x + 1}{x^2 - 9}\) is \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).

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