Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Domain of a Function Find the domain of the function. $$h(x)=\frac{x}{x^{2}-9}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{x}{x^{2}-9}\) is \(x \in \mathbb{R}\) such that \(x \neq -3, 3\).

Step 1: Set the Denominator Equal to Zero

The first step involves setting the denominator equal to zero and solve for \(x\). The equation is \(x^{2}-9 = 0\). Here, we add 9 to both sides, and then take square root on both sides. The square root of 9 is 3. So, \(x^{2} = 9\) implies \(x = \pm 3\). This means that for \(x = 3\) and \(x = -3\), the denominator of the function is zero and the function is undefined.

Step 2: Determine the Domain of the Function

The domain of the function \(h(x)=\frac{x}{x^{2}-9}\) is all real numbers except for the values for which the function is undefined. From the previous step, we found that the function is undefined for \(x = 3\) and \(x = -3\). Hence, the domain of the function is all real numbers excluding -3 and 3.

Key Concepts

Rational Functions

Rational functions are a type of function that can be expressed as the quotient of two polynomials. This means they are written in the form \( f(x) = \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
Rational functions are important because they help describe a wide variety of real-world situations and exhibit interesting behaviors, particularly where the denominator is zero, as these are points or lines of discontinuity.

• Numerator \( P(x) \) determines the zeros or roots of the function, indicating where the function itself is zero.
• Denominator \( Q(x) \) is especially crucial, as it determines the points at which the function is undefined, known as undefined points or discontinuities.

In the context of our function \( h(x) = \frac{x}{x^2 - 9} \), the numerator \( P(x) = x \) is a linear polynomial, while the denominator \( Q(x) = x^2 - 9 \) is a quadratic polynomial. When solving for the domain, ensuring the denominator does not equal zero is a key step.

Undefined Points

In any function, undefined points are values of \( x \) for which the function does not produce a valid output. For rational functions, this typically happens when the denominator is zero.
Remember, dividing by zero is mathematically undefined, and this condition leads to undefined points or discontinuities in your function. To find where a rational function like \( h(x) = \frac{x}{x^2 - 9} \) is undefined, we set the denominator equal to zero and solve for \( x \).

• For \( x^2 - 9 = 0 \), we add 9 to both sides to get \( x^2 = 9 \).
• Taking the square root of both sides gives \( x = \pm 3 \).

Thus, the function \( h(x) \) is undefined at \( x = 3 \) and \( x = -3 \). Understanding undefined points helps us understand the domain and behavior of rational functions and can be visualized as vertical asymptotes on a graph.

Real Numbers Exclusion

Determining the domain of a function is about identifying all possible inputs \( x \) that will produce a valid output from the function. For rational functions, this involves excluding values that cause the denominator to be zero, thus creating undefined points. The domain is generally all real numbers except those that make the function undefined.
Consider \( h(x) = \frac{x}{x^2 - 9} \). We already determined the function is undefined at \( x = 3 \) and \( x = -3 \).

• Therefore, we exclude 3 and -3 from the set of real numbers when expressing the domain.
• This can be denoted as: \( \{ x \in \mathbb{R} \mid x eq 3, x eq -3 \} \).

Understanding real numbers exclusion formalizes how gaps in the domain arise in rational functions, ensuring these functions are only evaluated at points where they produce valid numerical output. Excluding these undefined values maintains the integrity of calculations and analysis. Understanding these concepts is crucial for mastering function domains and rational expressions.