Find (f o g)(x) and its domain when f(x) = x^2+2 and g(x)= sqrt x+5
To find (f o g)(x), we substitute g(x) into f(x):
(f o g)(x) = f(g(x)) = (sqrt(x) + 5)^2 + 2.
The domain of (f o g)(x) will be the same as the domain of g(x) since we are substituting g(x) into f(x).
The domain of g(x) = sqrt(x) + 5 is restricted to x ≥ -5 since we cannot take the square root of a negative number.
Therefore, the domain of (f o g)(x) is also x ≥ -5.
To find (f o g)(x), we need to substitute g(x) into f(x) and simplify.
First, let's find g(x):
g(x) = sqrt(x+5)
Next, let's substitute g(x) into f(x):
f(g(x)) = f(sqrt(x+5))
Now, substitute the expression for g(x) into f(g(x)):
f(g(x)) = f(sqrt(x+5)) = (sqrt(x+5))^2 + 2
Simplifying, we have:
f(g(x)) = (x+5) + 2
f(g(x)) = x + 7
Therefore, (f o g)(x) = x + 7.
The domain of (f o g)(x) is the same as the domain of g(x), which is all real numbers such that x+5 ≥ 0 since we cannot take the square root of a negative number.
So, the domain of (f o g)(x) is x ≥ -5.
To find (f o g)(x), we need to substitute g(x) into f(x) and simplify.
Step 1: Start with g(x)
g(x) = sqrt(x) + 5
Step 2: Substitute g(x) into f(x)
f(g(x)) = f(sqrt(x) + 5)
Step 3: Substitute sqrt(x) + 5 into f(x)
f(g(x)) = (sqrt(x) + 5)^2 + 2
Step 4: Expand and simplify
f(g(x)) = (x + 10sqrt(x) + 25) + 2
f(g(x)) = x + 10sqrt(x) + 27
Therefore, (f o g)(x) = x + 10sqrt(x) + 27.
Now let's find the domain of (f o g)(x).
The domain of a composition of functions is the set of all x-values for which both g(x) and f(g(x)) are defined.
In this case, g(x) = sqrt(x) + 5 is defined as long as the expression inside the square root is non-negative:
x ≥ 0.
For f(g(x)) = x + 10sqrt(x) + 27 to be defined, the square root expression must also be non-negative:
sqrt(x) + 5 ≥ 0
sqrt(x) ≥ -5
Since the square root function is always non-negative, there is no additional restriction on the domain from f(g(x)). Therefore, the domain of (f o g)(x) is x ≥ 0.