Question

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

Answer 1

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

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

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.

Answer 2

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

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)