What is the range of $(f+g)(x)$ where $f(x)=x^2+4x-3$ and $g(x)=3x^2-8x+9$?

Question

What is the range of $(f+g)(x)$?

I plugged in the domain values to get the range for each of the equations and then I would have summed them up. But I'm getting erroneous results.

Because each function will have a different range, I believe we should first find these intervals and then get the sum.

The answer is given as $[5,30]$

How can I solve this problem?

Answer 1

The domain is $[-2,5)\cap(-4,1)=[-2,1)$, $$(f+g)(x)=f(x)+g(x)=4x^2-4x+6=(2x-1)^2+5\geq5.$$ The equality occurs for $x=\frac{1}{2}\in[-2,1).$

Now, calculate $(f+g)(-2)$, $(f+g)(1)$ and get the answer.

I got $[5,30]$.