Question

The symmetricity of the function f(x) = e^x + e^-x is about
a) x -axis b) y axis c) origin d) both a and c

Answer 1

Consider f(x)=e

x

+e

−x

Now f(−x)=e

−x

+e

x

=f(x)

Since

f(−x)=f(x), hence it is an even function.

Therefore it is symmetric about the y axis.

However f(x) intersects the y axis at y=1.

Hence the above f(x) is symmetric about the point (0,1).

Therefore it is not symmetric about the origin.

Now consider

f(x)=ln(x)

As x→0

f(x)→−∞

Furthermore the domain of f(x) is x>0.

The graph of f(x)=ln(x) decreases steeply in the region (0,1) as compared to the interval x>0.

Hence the graph f(x)=lnx is not symmetric about the origin.

Thirdly consider

f(x+y)=f(x)+f(y)

Then f(x) is of the form f(x)=λx where λ is a constant.

Now this represents a straight line passing through the origin, having a slope λ.

Hence it is symmetric about the origin.