Range question

Find the range of f (x) = cosx-sinx
Why is the answer -root2 < f (x) < root2

Shouldnt it be between 1 and -1 because sinx and cosx can take values between 1 and -1

Shouldnt it be between 1 and -1 because sinx and cosx can take values between 1 and -1

No. Put the function in R-alpha form and then it's obvious.

The function $f(x) = x$ can take values between $-\infty$ and $\infty$. Does that mean that $x-x$ should also takes values between $-\infty$ and $\infty$?

Thanks both. Can you also answer a question related to domain.
I can't link it as I'm on this on my phone but on June 2012 aea paper question 1c it asks to find the domain and I don't know why it's not X can be any real value do you combine both domains then take the wider one?

The domain is the admissible values that you can feed into $gf$, but when you feed a number into $gf$, you feed it into $f$ first and the domain of $f$ is $x\geqslant 0$, this will always produce an output $f(x) \geqslant 5$ so you can safely feed it into $g$ and you have the domain of $gf$ to be $x \geqslant 0$.

This answer would change if the range of $f$ didn't match up with the domain of $g$, say you had a value $f(x) = 1$, then you couldn't feed $f(x) = 1$ into $g$ because the domain of $g$ is $x\geqslant 2$, so it cannot accept $1$ as an input. you would then have to restrict the domain of $f$ (and consequently $gf$) so that the restriction of $f$ had a range which is a subset of the domain of $g$, fortunately you need to do no such thing here because of the first paragraph.