Suppose g is an even function and let h = f.g. Is h always an even function?

Solution:

A function is even if f(x) = f(−x) for all x.

This means that the function is the same for +ve x-axis and ,-ve x-axis, or graphically, symmetric about the y-axis.

Given:

g is an even function and h = f.g

The even function is defined as function that even if we replace ‘x’ with ‘-x’ we get the same function

Let us consider an example of g being an even function

Let g = x² and h = f(g(x))

⇒ h = f(x²)

f(x²) will always be positive irrespective of type of function as we have x² in it which always gives a positive value.

Hence,

if g is an even function and h = f.g then h will always be even function

Summary:

If g is an even function and h = f.g then h always an even function.