Questions: HW 3.5: Intro to Rational Functions Question 2 of 11 (9 points) I Question Attempt: 1 of 3 Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. l(x) = (3x-4) / (15x^2 - 2x - 8) Part 1 of 3 l(x) = (3x-4) / (15x^2 - 2x - 8) will not be a real number when the denominator is zero. Part: 1 / 3 Part 2 of 3 That is, when 15x^2 - 2x - 8 = 0 ( )( ) = 0 x = and x =

HW 3.5: Intro to Rational Functions Question 2 of 11 (9 points) I Question Attempt: 1 of 3 Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. l(x) = (3x-4) / (15x^2 - 2x - 8) Part 1 of 3 l(x) = (3x-4) / (15x^2 - 2x - 8) will not be a real number when the denominator is zero. Part: 1 / 3 Part 2 of 3 That is, when 15x^2 - 2x - 8 = 0 ( )( ) = 0 x = and x =
Transcript text: HW 3.5: Intro to Rational Functions Question 2 of 11 (9 points) I Question Attempt: 1 of 3 Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. \[ l(x)=\frac{3 x-4}{15 x^{2}-2 x-8} \] Part 1 of 3 $l(x)=\frac{3 x-4}{15 x^{2}-2 x-8}$ will not be a real number when the denominator is zero. Part: 1 / 3 Part 2 of 3 That is, when \[ \begin{aligned} 15 x^{2}-2 x-8 & =0 \\ (\square)(\square) & =0 \\ x & =\square \end{aligned} \] and $x=$ $\square$
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Solution

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Solution Steps

To find the domain of the function \( l(x) = \frac{3x-4}{15x^2 - 2x - 8} \), we need to determine where the denominator is zero, as the function is undefined at these points. This involves solving the quadratic equation \( 15x^2 - 2x - 8 = 0 \). The solutions to this equation will be the values of \( x \) that are excluded from the domain. We can use the quadratic formula to find these solutions.

Solution Approach
  1. Identify the quadratic equation in the denominator.
  2. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.
  3. Exclude these roots from the domain.
  4. Express the domain in interval notation.
Step 1: Identify the Quadratic Equation

The function \( l(x) = \frac{3x-4}{15x^2 - 2x - 8} \) has a denominator given by the quadratic equation: \[ 15x^2 - 2x - 8 = 0 \]

Step 2: Solve the Quadratic Equation

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find the roots of the equation. Here, \( a = 15 \), \( b = -2 \), and \( c = -8 \). The solutions to the equation are: \[ x = -\frac{2}{3} \quad \text{and} \quad x = \frac{4}{5} \]

Step 3: Determine the Domain

The function \( l(x) \) is undefined at the points where the denominator is zero, which are the solutions found above. Therefore, the domain of \( l(x) \) excludes these values. In interval notation, the domain is expressed as: \[ (-\infty, -\frac{2}{3}) \cup (-\frac{2}{3}, \frac{4}{5}) \cup (\frac{4}{5}, \infty) \]

Final Answer

The domain of the function \( l(x) \) in interval notation is: \[ \boxed{(-\infty, -\frac{2}{3}) \cup (-\frac{2}{3}, \frac{4}{5}) \cup (\frac{4}{5}, \infty)} \]

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