Question

What is the domain and range of this function?

Answer 1

As for the domain, the only restriction comes from the logarithm. The argument of a logarith must be strictly positive, so we have

$x-2>0 \iff x>2$

As for the range, we have:

• The range of $\log(x)$ are all real numbers
• If we change to $\log(x-2)$ we're translating the function horizontally, so the range remains the same
• If we change to $\log(5(x-2()$ we're stretching the function horizontally, so the range doesn't change
• If we change to $\log(5(x-2))+1$ we're translating the function 1 unit up, but the range is already all the real numbers, so it doesn't change.