Given the equations $y=3\sin x+2$ and $y=x+c$, which statements are true?

Question

$y=3\sin x+2$

$y=x+c$ where c is a constant

Which of the following statements is/are true?

1. For some value of c: there is exactly one solution with $0\leq x\leq \pi$ and there is at least one solution with $-\pi< x<0$

2. For some value of c: there is exactly one solution with $0\leq x\leq \pi$ and there are no solutions with $-\pi< x<0$

3. For some value of c: there is exactly one solution with $0\leq x\leq \pi$ and there are no solutions with $x>\pi$

And why?

I tried solving it by picking values of x and seeing if they fit the statements.

Answer 1

If you graph the functions (should be easy to do by hand. If you don't know how to graph the sine function, learn here), a solution is where the graphs intersect.

Where c = 0:

There is one solution between 0 and $\pi$. This proves option 3 and option 2 correct.

Where c = 1:

There are 3 solutions, two between $-\pi$ and 0 and one between 0 and $\pi$. This proves option 1 to be correct.

There you have it, all of the options are correct for the given functions.