Question

Question #7a8d5

Answer 1

`D = -oo < x < 2 | x in RR`
`R = -oo < y < oo | y in RR`

Explanation:

The range is every value of `y` possible, or, in other words, every value that `f(x)` can be whereas the domain is every value of `x` we can plug in without giving a math error (division by zero, even root of a negative number, log of a null or negative number).

As for range, a logarithmic function such as the one presented can take every value between minus infinity and infinity.

You can either look at the graph, you can plug in arbitrary values and see how the function behaves, or think about it conceptually, since every number between 0 and 1 (non inclusive) has a unique negative logarithm, and every number between 1 and infinity has a unique positive or null logarithm and there are infinite numbers in those range, then the function must span from minus infinity to infinity. So range `R = -oo < y < oo | y in RR`

Now, logarithmic functions can only take non-zero, positive arguments, so we must make sure that `-x+2 > 0`

Which is an easy inequality to solve
`-x + 2 > 0 rarr 2 > x`

So the domain `D = -oo < x < 2 | x in RR`

Also, protip, to write a log of any base write log_k where k is the base. The underline makes stuff subscript, although the base e logarithm is often written as ln for shorthand (because it's the natural logarithm).