Problem 40 - Elementary and Intermediate Algebra

Question

Find the domain of \(f\) $$ f(x)=\frac{3 x}{x^{2}-1} $$

Step-by-Step Solution

Answer

The domain of \(f(x)\) is all real numbers except \(x = 1\) and \(x = -1\).

Step 1 - Identify Denominator

Determine the expression in the denominator of the function. In this case, the denominator is: \[x^2 - 1\]

Step 2 - Set Denominator Not Equal to Zero

For the function to be defined, the denominator must not be zero. Set the denominator equal to zero and solve for \(x\): \[x^2 - 1 = 0\]

Step 3 - Solve the Equation

Solve the equation \(x^2 - 1 = 0\) to find the values of \(x\) that make the denominator zero: \[x^2 - 1 = (x - 1)(x + 1) = 0\]\[x - 1 = 0 \quad \text{or} \quad x + 1 = 0\]\[x = 1 \quad \text{or} \quad x = -1\]

Step 4 - Determine the Domain

The domain includes all real numbers except where the denominator is zero. From the solutions in Step 3, the function is undefined at \(x = 1\) and \(x = -1\). Therefore, the domain of \(f(x)\) is all real numbers except \(x = 1\) and \(x = -1\): \[\text{Domain of } f: \{x \, | \, x eq 1 \, \text{and} \, x eq -1\}\]

Key Concepts

Rational Functions

Rational functions are a type of function represented as the ratio of two polynomials. The general form of a rational function is given by:
\[ f(x) = \frac{P(x)}{Q(x)} \].
Here, \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) e 0\), as dividing by zero is undefined.
In the given example, the function is \(f(x) = \frac{3x}{x^2 - 1}\), where \(3x\) is the numerator \(P(x)\), and \(x^2 - 1\) is the denominator \(Q(x)\). It’s crucial to understand the behavior of both polynomials to determine where the function is valid, particularly focusing on the points that might make the function undefined.

Denominator

The denominator of a rational function plays a key role in defining its domain. To find where the function is defined, we need to ensure the denominator is never zero.
In our function \(f(x) = \frac{3x}{x^2 - 1}\), \(x^2 - 1\) is the denominator.
To find the values that make the denominator zero, we set the denominator equal to zero and solve for \(x\).
If \(x^2 - 1 = 0\), it can be factored into \((x - 1)(x + 1) = 0\). This equation tells us that the values of \(x\) making the denominator zero are \(x = 1\) and \(x = -1\). Thus, the function \(f(x)\) is undefined at these points.
Recognizing these critical points helps us identify the domain of the function by excluding them.

Solving Equations

Solving equations is a fundamental skill for understanding and manipulating functions. In defining the domain of a rational function, solving equations helps pinpoint where the denominator becomes zero.
>In the equation \(x^2 - 1 = 0\), we solved:

• First, factorize it to \((x - 1)(x + 1) = 0\).


• Then, set each factor to zero: \(x - 1 = 0\) and \(x + 1 = 0\).


• Solving these separately, we find \(x = 1\) and \(x = -1\).

These points cause the denominator to be zero, making the function undefined at these values. To determine the domain, we exclude \(x = 1\) and \(x = -1\) from all real numbers, ensuring the function is defined.