Question

question 5 of 25
consider the function.
f(x) = \frac{x - 7}{5x - 4}
find the domain for the function in interval notation.
\bigcirc (-\infty, 4 / 5 \cup 4 / 5, \infty)
\bigcirc -\infty, 4 / 5 \cup 4 / 5, \infty
\bigcirc (-\infty, 4 / 5) \cup (4 / 5, \infty)
\bigcirc -\infty, 4 / 5) \cup (4 / 5, \infty

Explanation:

Step1: Identify the denominator

The function is a rational function \( f(x) = \frac{x - 7}{5x - 4} \). For a rational function, the denominator cannot be zero. So we set the denominator equal to zero and solve for \( x \):
\( 5x - 4 = 0 \)

Step2: Solve for \( x \)

Adding 4 to both sides: \( 5x = 4 \)
Dividing both sides by 5: \( x = \frac{4}{5} \)

Step3: Determine the domain

The domain of a rational function is all real numbers except the value that makes the denominator zero. So we exclude \( x = \frac{4}{5} \). In interval notation, this is \( (-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty) \).

Answer:

\( (-\infty, 4/5) \cup (4/5, \infty) \) (the third option)