Question

at what points is the function ( y = \frac{sin x}{6x - 6} ) continuous? describe the set of x-values where the function is continuous, using interval notation. (simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Find where the denominator is zero

To determine the points of discontinuity, we first find the values of \( x \) that make the denominator \( 6x - 6 = 0 \).
Solve the equation \( 6x - 6 = 0 \):
Add 6 to both sides: \( 6x = 6 \)
Divide both sides by 6: \( x = 1 \)

Step2: Determine the domain of continuity

The function \( y=\frac{\sin x}{6x - 6} \) is a rational function (a quotient of two functions). The numerator \( \sin x \) is continuous for all real numbers, and the denominator is a polynomial, which is also continuous for all real numbers. However, the rational function is undefined (and thus discontinuous) where the denominator is zero, i.e., at \( x = 1 \).
So the function is continuous for all real numbers except \( x = 1 \). In interval notation, this is \( (-\infty, 1) \cup (1, \infty) \)

Answer:

\( (-\infty, 1) \cup (1, \infty) \)