The period of the function $f(x)=a\cdot \sin(ax)+a$

Question

What is the period of the following function $$f(x)=a\cdot \sin(ax)+a, \mbox{ } x \in \mathbb{R}, a>0.$$ How can I find out?

Thanks.

Answer 1

it should be such that $$f(x)=f(x+t)$$ where t is period. Then since $$\sin(ax)=\sin(ax+at)$$, we get $t=\frac{2\pi}{a}$.

Answer 2

since period of $\sin x$ is $2\pi$ then period of $\sin ax$ is $\frac{2\pi}{a}$

Answer 3

Assume the period is $c$ and use the condition for periodicity of a function

$$ f(x+c)=f(x). $$

Work out the above equation and see what you get.

Answer 4

If $g(x)$ has period $p$ and $u,v$ are nonzero then $ug(vx)+w$ has period $\frac p{|v|}$.