Quick Integration Approximation Question

Question

So I was given the following question:

"The continuous function $f$ is defined on the interval $-5 \leq x \leq8$. The graph of $f$, which consists of four line segments, is shown in the figure above. Let $g$ be the function given by $g(x)=2x+\int_2^1f(t)dt$. Find $g(0)$ and $g(-5)$."

I'm a bit confused on how I'd go about tackling something like this. Would I literally just take the given values of $f(x)$ on the graph and use then in finding the values that were asked for? Any help would be appreciated!

Answer 1

Yeah. I think that if they give you a picture, then it might be because they want you to extrapolate data from it (also notice that it gives you the scale of the plane). But you can just use the marked points as a reference to reconstruct $f(x)$ using the equation of a line given two points.

Would I literally just take the given values of $f(x)$ on the graph and use then in finding the values that were asked for?

In that sense, you're right. You would have to integrate $f(x)$ within the given bounds by first reconstructing it. This is how it should look like:

\begin{equation} f(x)=\begin{cases}2x+6, &\text{ if }&-5\leq x< -2;\\-\frac{1}{2}x+1, &\text{ if }&-2\leq x <0;\\x+1, &\text{ if }&0\leq x < 2; \\ -x+5, &\text{ if }&2\leq x\leq 8.\end{cases} \end{equation}

However, you could just reconstruct the function within the integration bounds.

Note that the previous part implies that:

\begin{equation}\int_{2}^{1}f(x) \ dx=-\int_1^2f(x) \ dx=-\int_1^2 (x+1)\ dx = (?).\end{equation}

Finally, you have to evaluate $g(x)$ at the given values of $x$.