Identify the Domain of each function. Use interval notation

Question

f(x)=log(6x-7)/x²-8x

Answer 1

You can only take the log of a positive number and a numerator can't equal 0.

6x - 7 > 0

6x > 7

x > 7/6

x² - 8x = 0

x(x - 8) = 0

One of the factors, x or x - 8 has to equal 0 for the product to equal 0.

x = 0

x - 8 = 0

x = 8

So, our input has to be greater than 7/6. We can't use 0 because it would make the denominator 0 aaaaand it's less than 7/6, and we can't use 8 because it would make the denominator 0.

(7/6,8) U (8,∞)

Answer 2

[Note: In answering this question, I am assuming that the denominator of the fraction is x²-8x, and not just the x².]

The domain of a function is basically the numbers that we can plug into the function and have it still make sense (the numbers we can plug in without breaking math).

In the numerator, we have a logarithm. The input for a logarithm always needs to be positive. This means that 6x-7 needs to be positive, which means that it needs to be greater than 0.

6x-7>0

6x>7

x>7/6

In the denominator, we have x²-8x. Denominators can never be equal to 0.

x²-8x ≠ 0

x(x-8) ≠ 0

x ≠ 0 AND x-8 ≠ 0

x ≠ 0 AND x ≠ 8

Altogether, we know that x>7/6, and x cannot be 0 or 8.

Since x>7/6, and 7/6 is bigger than 0, we don't need to worry about how x can't be 0, because anything bigger than 7/6 already won't be 0.

x>7/6, and x ≠ 8

Start the interval with the smallest number, 7/6. Since x has to be bigger than 7/6, can cannot be equal to 7/6, we need an open parentheses ( instead of [. We'll go up until we reach a number that isn't included, which in this case is 8. That will also be a ) instead of ] because x cannot be 8.

(7/6, 8)

Then we need to include numbers bigger than 8, since those are still bigger than 7/6 and are not equal to 8. Everything bigger than 8 works, so the next interval will go to infinity. Again, use open parentheses.

(8, ∞)

Put both intervals together with the word "and" or with the symbol ∪.

The domain is (7/6, 8) ∪ (8, ∞)