Questions: Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. k(x)=(5x-3)/(3x^2-2x-5)

Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. k(x)=(5x-3)/(3x^2-2x-5)
Transcript text: Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. \[ k(x)=\frac{5 x-3}{3 x^{2}-2 x-5} \]
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Solution

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Solution Steps

To find the domain of the function \( k(x) = \frac{5x-3}{3x^2-2x-5} \), we need to determine where the denominator is not equal to zero, as division by zero is undefined. This involves solving the equation \( 3x^2 - 2x - 5 = 0 \) to find the values of \( x \) that make the denominator zero. The domain will be all real numbers except these values.

Step 1: Identify the Denominator

The function is given by

\[ k(x) = \frac{5x - 3}{3x^2 - 2x - 5} \]

To find the domain, we need to analyze the denominator:

\[ 3x^2 - 2x - 5 \]

Step 2: Solve for Zeros of the Denominator

We set the denominator equal to zero to find the values of \( x \) that make the function undefined:

\[ 3x^2 - 2x - 5 = 0 \]

The solutions to this equation are:

\[ x = -1 \quad \text{and} \quad x = \frac{5}{3} \]

Step 3: Determine the Domain

The function \( k(x) \) is undefined at the points \( x = -1 \) and \( x = \frac{5}{3} \). Therefore, the domain of \( k(x) \) in interval notation is:

\[ (-\infty, -1) \cup (-1, \frac{5}{3}) \cup (\frac{5}{3}, \infty) \]

Final Answer

The domain of the function \( k(x) \) is

\[ \boxed{(-\infty, -1) \cup (-1, \frac{5}{3}) \cup (\frac{5}{3}, \infty)} \]

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