Find a function $g(x)$ satisfying the above conditions.

Question

Find a function $g(x)$ satisfying the above conditions:-

a)domain is $(-∞,∞)$.

b)range is $[-2,8]$.

c)$g(x)$ has a period $π$.

d)$g(2)$=3.

ATTEMPT:

Since the function is periodic with period $π$ it must be a trigonometric function, and since range is $[-2,8]$, it can be of the form $ a + \sin(2x)$.

Answer 1

Hint: It should be of the form $a+b\sin(2x)$. How would you figure out the value of $a$ and $b$? Consider the range of $\sin(x)$.

More Hint: The range of $\sin(x)$ is symmetric. It looks like $[-y,y]$. Yet the range you want is not symmetric. How should we fix this?

Consider $3+5\sin(2x-4)$.

Answer 2

Take the sin curve and look at its range. As you want a function in between $[-2,8]$ Tweak the function a bit.

Think about what will happen in $f(x)=x$ when I add 1 to the RHS ,what happens when I multiply 5 to the RHS. Think about the graph.