Can we identify the right sketch of a product of functions?

Can we identify the right sketch of a product of functions?

Question

On which of the axes below is a sketch of the graph $y = 2^{-x} \sin^2 (x^2)?$

Suggestion

When are the functions $2^{-x}$ and $\sin^2 (x^2)$ positive, and when are they negative??

What is the $y$ coordinate of the graph in the question when $x=0$?

When does the graph in the question meet the $x$-axis?

Solution

• Both $2^{-x}$ and $\sin^2(x^2)$ are functions that are never negative, so $y\geq0$, which discounts b.
• At $x=0$, $2^{-x} = 1$ and $\sin^2 (x^2)=0$, so $y(0)=0$, which discounts d.
• Finally, the graph meets the $x$-axis when $\sin^2 (x^2) = 0$, which occurs when $\sin (x^2)=0$, so at $x = \sqrt{\pi}, \sqrt{2\pi}, \sqrt{3\pi}, \ldots.$ These are not evenly spaced as in c. These values get closer together as $x$ increases, as in a.

Hence the answer is (a).