Find (f ∘g)(x) and its domain when f (x) = x^2 + 2 and g (x) = sqrt x+5
To find (f ∘ g)(x), we substitute g(x) into f(x).
First, we find g(x):
g(x) = √(x + 5)
Next, we substitute g(x) into f(x):
f(g(x)) = f(√(x + 5))
Now, we substitute g(x) = √(x + 5) into f(x) = x^2 + 2:
(f ∘ g)(x) = (√(x + 5))^2 + 2 = (x + 5) + 2 = x + 7
Therefore, (f ∘ g)(x) = x + 7.
Now, let's find the domain of (f ∘ g)(x). Since (f ∘ g)(x) = x + 7 is a simple linear function, its domain is all real numbers, which means the domain is (-∞, ∞).
wrong, since the domain of g is x >= -5
Apologies for the mistake. You are correct.
To find (f ∘ g)(x), we substitute g(x) into f(x).
First, we find g(x):
g(x) = √(x + 5)
Next, we substitute g(x) into f(x):
f(g(x)) = f(√(x + 5))
Now, we substitute g(x) = √(x + 5) into f(x) = x^2 + 2:
(f ∘ g)(x) = (√(x + 5))^2 + 2 = (x + 5) + 2 = x + 7
Therefore, (f ∘ g)(x) = x + 7.
Now, let's find the domain of (f ∘ g)(x). Since the domain of g(x) is x >= -5, the values of x that we can substitute into (f ∘ g)(x) are restricted to x >= -5.
Thus, the domain of (f ∘ g)(x) is x >= -5.