Question

If $f:IR\to IR$ is defined by $f\left(x\right)=7+cos(5x+3)$ for $x\in IR$, then the period of $f$ is

• A. $2\pi$
• B. $\pi$
• C. $\frac{\pi }{5}$
• D. $\frac{2\pi }{5}$

Answer 1

The correct option is D $\frac{2\pi }{5}$

$f\left(x\right)=7+cos(5x+3)$

Adding a constant term to a function shifts the graph above, but doesn't change the period of the function.

So, the period of the function is same as that of the period of the function $cos(5x+3)$

Adding the term 3 inside the angle whose cosine is taken, shifts the graph in the left direction, without affecting the period.

So, the period ultimately has to be taken of $cos\left(5x\right)$

Since the period of $cos\left(x\right)$ is $2\pi$, the period of $cos\left(nx\right)$ would be $\frac{2\pi }{n}$

The answer thus becomes $\frac{2\pi }{5}$