Finding range of transformation of function from range of original

Question

I'm asked to find the range of $y = f(x-2)+4$, if the range of $y=f(x)$ is
{$y| -2 \geq y \geq 5, y \in R$}.
How do I go about finding this? I have no idea where to even start. I'm doing the course online and the explanations were not very adequate, that's why I have made literally zero progress on this problem. I just need some helpful suggestions to get started.
I appreciate the help.

Answer 1

The range of a function is the set of outputs. In other words, the range of $f$ is the interval $[-2, 5]$ of real numbers, then

• for any $y$ in that interval (i.e. $-2 \le y \le 5$), there exists some $x$ (possibly more than one) such that $f(x) = y$, and
• for any $y$ not in that interval (i.e. $y < -2$ or $y > 5$), there are no numbers $x$ with $f(x) = y$.

Now you are considering a new function. Let's give it a name: $g$. It is defined in terms of $f$: $$ g(x) = f(x - 2) + 4 $$ Any output of $f$ is going to produce an output of $g$ simply by adding $4$ to it.

To make this concrete, say that $f(7) = 1$. (We know that some number $x$ makes $f(x) = 1$ since $1$ is in the range of $f$). Now, calculate $$ \begin{align} g(9) &= f(9 - 2) + 4 \\ &= f(7) + 4 \\ &= 1 + 4 \\ &= 5 \end{align} $$ This shows that $5$ is in the range of $g$. Can you figure out exactly which numbers are possible outputs of $g$? Hint: it's an interval, just as it is for $f$.