State the inner and outer function of f (x) = arctan (e^ (x))

The inner function is v (x) = e^x

and the outer function is f (x) = arctan (e^x)

Step-by-step explanation:

Given f (x) = arctan (e^x)

Let v = e^x, and f (x) = y

Then y = arctan (v)

This implies that y is a function of u, and u is a function of x.

Something like y = f (v) and v = v (x)

y = f (v (x))

This defines a composite function.

Here, v is the inner function, and arctan (u) is the outer function.

Since v = e^x, we say e^x is the inner function, and arctan (e^x) is the outer function.

So the inner function is g (x) = arctan (x) and the outer function is h (x) = e^ (x)

Step-by-step explanation:

Suppose we have a function in the following format:

f (x) = g (h (x))

The inner function is h (x) and the outer function is g (x).

In this question:

f (x) = arctan (e^ (x))

From the notation above

h (x) = e^ (x)

g (x) = arctan (x)

Then

f (x) = g (h (x)) = g (e^ (x)) = arctan (e^ (x))

So the inner function is g (x) = arctan (x) and the outer function is h (x) = e^ (x)