GATE Overflow Test Series | Spatial and Analytical Aptitude | Test 1 | Question: 20

Question

For which of the following function(s), reflection about the $y-$axis give(s) the same function? (Mark all the appropriate choices)

• A. $f(x) = \sin x$
• B. $f(x) = \cos x$
• C. $f(x) = \sec x$
• D. $f(x) = \csc x$

Answer 1

While reflecting a point about the $y$-axis, the magnitude of its $x$-coordinate remains the same but its sign changes. But the $y$-coordinate remains the same. This is because we are taking it to the quadrant beside it. So, we can say that ${\text{R}}_{y}(x,y) = (-x,y)$.

Even functions are functions that satisfy $f(x) = f(-x)$ for all $x.$ Even functions are symmetric about the line $x =0$ (means $y$-axis).

Odd functions are functions that satisfy $f(x) = -f(-x)$ for all $x.$ Odd functions exhibit point symmetry about the origin.

We know that,

• $\sin(-x) = -\sin x$ (odd function)
• $\cos(-x) = \cos x$ (even function)
• $\sec(-x) = \sec x$ (even function)
• $\csc(-x) = -\csc x$ (odd function)

$\therefore$ Clearly the function should be even.

So, the correct answer is $B;C.$