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Question

Let
f(x) = (x − 6)/(x − 1)(x + 5)
.

Use interval notation to indicate where f(x) is continuous.

Interval(s) of Continuity: ?

Answer 1

This function becomes discontinuous when denominator is zero. That is when x = -5 or x = 1. The function is continuous everywhere except those two points. Therefore intervals of continuity can be written as

(-oo,-5) U (-5,1) U (1, +oo)

Answer 2

This function becomes discontinuous when denominator is zero. That is when x = -5 or x = 1. The function is continuous everywhere except those two points. Therefore intervals of continuity can be written as

(-oo,-5) U (-5,1) U (1, +oo)

Answer 3

Hi,
The function becomes discontinuous when either term in the denominator becomes zero because zero multiplied by the other term is zero, and division by zero is not defined.
The usual way to solve for the number that makes an expression zero is to set it equal to zero and solve for x.
x-1 = 0
x = 1

x+5 = 0
x = -5

You will see that if you substitute 1 for x in x-1, that expression becmes zero, and likewise for -5 in x+5.

So, the function is discontinuous at:
x = 1 and x = -5. Notice that we don't care what the numerator is.

Now, we want to write this is interval notation:
(-oo, -5)U(-5,1)U(1, oo) Where I've used oo for infinity.

The parentheses mean that the term themselves are not included in the domain (If they had been we would have used brackets.), and the symbol "U" means union to indicate that all of those ranges of numbers are included.

Hope this helps.
FE