Question

$\text{For}x\in [-2\pi ,2\pi ],f\left(x\right)=\sin x;$
$g\left(x\right)=\cos x.$ Then, select the correct statements.

• A. $f(-x)\&-g\left(x\right)$ will have $4$ points of intersection
• B. $f\left(x\right)\&g\left(x\right)$ will have $4$ points of intersection
• C. $f(-x)\&g\left(x\right)$ will have $4$ points of intersection
• D. $f\left(x\right)\&-g\left(x\right)$ will have $4$ points of intersection

Answer 1

Answer: (A), (B), (C), (D)

$f\left(x\right)\&-g\left(x\right)$ will have $4$ points of intersection

Given: $x\in [-2\pi ,2\pi ],f\left(x\right)=\sin x\&g\left(x\right)=\cos x$
Now, the graph of $f\left(x\right)=\sin x$ can be drawn as:

Similarly, we can draw the graph of $g\left(x\right)=\cos x$ as:

Now, $f(-x)=\sin (-x),$ whose graph will be horizonatally flipped graph of $f\left(x\right)$ along the x- axis as:


Also, $g\left(x\right)=\cos x$
$\Rightarrow g(-x)=\cos (-x)=\cos x$
Hence, it's graph will be the same as the graph of $g\left(x\right)$
Now, $-g\left(x\right)=-\cos x$ whose graph will be the vertical flipped graph of $g\left(x\right)$ along the y-axis as:

Now that we have all the graphs, let's plot the graphs in the same plane and find out the number of points of intersection for $x\in [-2\pi ,2\pi ]$ for the following pair of functions.
$a.f\left(x\right)\&g\left(x\right)$ Or $\sin x\&\cos x:$


Thus from the graph there are $4$ points of intersection.

$b.f(-x)\&g\left(x\right)$ Or $\sin (-x)\&\cos x:$


Thus these two functions have 4 points of intersection as well.

$c.f\left(x\right)\&-g\left(x\right)$ Or $\sin x\&-\cos x$


Thus these two functions have 4 points of intersection as well.

$d.f(-x)\&-g\left(x\right)$ Or $\sin (-x)\&-\cos x$


Thus these two functions have 4 points of intersection as well.