Given the functions f(x) = 2^x + 1 and g(x) = x - 3, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
(f∘g)(x) = f(g(x)) = 2^g(x) + 1 = 2^(x-3) + 1
do the same for (g∘f)
To determine the equation for the composition of functions (f ∘ g)(x) and (g ∘ f)(x), we need to substitute the function g(x) into f(x) and vice versa.
Let's start with (f ∘ g)(x):
1. Start with the original function f(x) = 2^x + 1.
2. Substitute g(x) into f(x): f(g(x)) = 2^(g(x)) + 1.
3. Replace g(x) with its expression: f(g(x)) = 2^(x - 3) + 1.
4. Simplify if necessary: (f ∘ g)(x) = 2^(x - 3) + 1.
Now let's find the equation for (g ∘ f)(x):
1. Start with the original function g(x) = x - 3.
2. Substitute f(x) into g(x): g(f(x)) = f(x) - 3.
3. Replace f(x) with its expression: g(f(x)) = (2^x + 1) - 3.
4. Simplify if necessary: (g ∘ f)(x) = 2^x - 2.
So, the equation for (f ∘ g)(x) is 2^(x - 3) + 1, and the equation for (g ∘ f)(x) is 2^x - 2.