Question

at what points is the given function f(x) continuous?
$f(x)=\frac{6}{x-4}-5x$
describe the set of x-values where the function is continuous, using interval notation.
(use interval notation.)

Explanation:

Step1: Identify Discontinuity

The function \( f(x)=\frac{6}{x - 4}-5x \) has a rational part \( \frac{6}{x - 4} \). A rational function is undefined when its denominator is zero. So, set \( x - 4 = 0 \), which gives \( x = 4 \). This is the point of discontinuity.

Step2: Determine Continuous Intervals

For all real numbers except \( x = 4 \), the function is a combination of a rational function (continuous on its domain) and a polynomial (which is continuous everywhere). So the function is continuous on \( (-\infty, 4) \) and \( (4, \infty) \).

Answer:

\((-\infty, 4) \cup (4, \infty)\)