Question

Given f(x) = e^x and g(x) = x – 2, what is the range of (g*f)(x)

Answer 1

The range of (g * f)(x) is all real numbers (ℝ).

Step-by-step explanation:

Composing functions: We first need to understand that (g * f)(x) refers to the composition of g and f, where f is evaluated first and then its output is used as the input for g. So, (g * f)(x) = g(f(x)) = g(e^x).

Analyzing g(e^x): The function g(x) simply subtracts 2 from its input. Since e^x is a real number for all real values of x, g(e^x) will also be a real number for all real values of x.

Range of the composed function: Therefore, as g(e^x) can take any real value, the range of (g * f)(x) will be all possible outputs of g(e^x), which is all real numbers (ℝ).

In simpler terms, no matter what real number you input into f(x), the final output after applying g(x) will always be a real number. This is because the subtraction in g(x) doesn't restrict the output in any way.

Answer 2

For this case we have the following functions:

`f (x) = e ^ x\\g (x) = x-2`

By definition we have to:

`(f * g) (x) = f (x) * g (x)`

Substituting:

`(f * g) (x) = e ^ x (x-2)`

The range of the function is given by all values of "and" valid. That is, all the real numbers.

ANswer:

(-∞,∞)

Graphic attached