If $f(x)$ is a Borel function, is $f(x+a)$ also a Borel function?

Question

I've been recently covering Borel functions and wondering are they preserved by the following property.

If $f(x)$ is a Borel function then $f(x+a)$ is also a Borel function where $a$ is a real number.

It makes sense to me as if $f(x)$ is a Borel set $(-\infty ,b)$ in a set in the Borel algebra so $f(x+a)$ would have a similar set $(-\infty, b-a)$ in the Borel $\sigma$-algebra and therefore is a Borel function.

Any help would be great thanks

Answer 1

A composition of Borel functions is Borel, so $f$ composed with $g(x)=x+a$ is Borel, because both $f$ and $g$ are Borel.