Question

Let f(x)=2^x and g(x)=x-2. The graph of (f o g)(c) is shown below. What is the domain of (f o g)(x)?

Answer 1

The domain of the outer function is all real numbers, because f(x) is an exponential function.

The domain and range of g(x) are all real numbers.

So, the domain of the composition is again all real numbers, because there is no way that an output from g(x) will not be a valid input for f(x).

Answer 2

For this case we have the following functions:

$f (x) = 2 ^ x\\g (x) = x-2$

We must find $(f_ {o} g) (x):$

By definition we have to:

$(f_ {o} g) (x) = f [g (x)]$

So:

$(f_ {o} g) (x) = 2 ^ {x-2}$

By definition, the domain of a function is given by all the values for which the function is defined. Thus, the domain of the composite function is:

In this case, there are no real numbers that make the expression indefinite.

Thus, the domain is given by all the real numbers.

Answer:

All the real numbers