Question

Let F(x) = e^x and g(x) = x - 3. What are the domain and range of (Fog)(x)?

A. Domain: x > 0, Range: y < 0
B. Domain: x > 3, Range: y > 0
C. Domain: all real numbers, Range: y < 0
D. Domain: all real numbers, Range: y > 0

Answer 1

The domain of (Fog)(x) is all real numbers and the range is all positive real numbers.

Step-by-step explanation:

The composition of two functions, (Fog)(x), is found by first applying the function g(x) to x and then applying the function f(x) to the result. In this case, g(x) = x - 3 and f(x) = e^x. Therefore, (Fog)(x) = f(g(x)).

To find the domain of (Fog)(x), we need to find the values of x for which the composition is defined. Since g(x) = x - 3 can take any real value, the domain of (Fog)(x) is the same as the domain of f(x), which is all real numbers.

To find the range of (Fog)(x), we need to find the values of y that can be obtained by applying the composition to different x values. In this case, the range of f(x) is all positive real numbers, so the range of (Fog)(x) is also all positive real numbers.