Let f(x)=x^2+2x-4 and g(x)= 1/x+1
A) g(f(0))
B) f(g(-2))
A) To find g(f(0)), we first need to evaluate f(0).
f(x) = x^2 + 2x - 4
f(0) = (0)^2 + 2(0) - 4
f(0) = 0 + 0 - 4
f(0) = -4
Now, we can substitute -4 into g(x).
g(x) = 1/(x+1)
g(f(0)) = g(-4+1)
g(f(0)) = g(-3)
Therefore, g(f(0)) = 1/(-3+1) = 1/(-2) = -1/2
B) To find f(g(-2)), we first need to evaluate g(-2).
g(x) = 1/(x+1)
g(-2) = 1/(-2+1)
g(-2) = 1/(-1)
g(-2) = -1
Now, we can substitute -1 into f(x).
f(x) = x^2 + 2x - 4
f(g(-2)) = f(-1)
f(g(-2)) = (-1)^2 + 2(-1) - 4
f(g(-2)) = 1 - 2 - 4
f(g(-2)) = -5
Therefore, f(g(-2)) = -5
To find the value of g(f(0)), we need to substitute 0 into the function f(x) first, and then substitute the result into the function g(x).
A) g(f(0)):
Step 1: Substitute 0 into f(x): f(0) = (0)^2 + 2(0) - 4 = 0 + 0 - 4 = -4
Step 2: Substitute the result (-4) into g(x): g(-4) = 1/(-4) + 1 = -1/4 + 1 = 3/4
Therefore, g(f(0)) = 3/4.
To find the value of f(g(-2)), we need to substitute -2 into the function g(x) first, and then substitute the result into the function f(x).
B) f(g(-2)):
Step 1: Substitute -2 into g(x): g(-2) = 1/(-2) + 1 = -1/2 + 1 = 1/2
Step 2: Substitute the result (1/2) into f(x): f(1/2) = (1/2)^2 + 2(1/2) - 4 = 1/4 + 1 - 4 = -15/4
Therefore, f(g(-2)) = -15/4.
To find the values of g(f(0)) and f(g(-2)), we need to substitute the given values into the functions and evaluate the resulting expressions.
A) To find g(f(0)), we'll start by calculating f(0) and then substitute the result into g(x).
1. Calculate f(0):
f(x) = x^2 + 2x - 4
f(0) = (0)^2 + 2(0) - 4
= 0 + 0 - 4
= -4
2. Substitute f(0) into g(x):
g(x) = 1/x + 1
g(f(0)) = g(-4) (since f(0) = -4)
= 1/(-4) + 1
= -1/4 + 1
Therefore, g(f(0)) = -1/4 + 1 = 3/4.
B) To find f(g(-2)), we'll start by calculating g(-2) and then substitute the result into f(x).
1. Calculate g(-2):
g(x) = 1/x + 1
g(-2) = 1/(-2) + 1
= -1/2 + 1
= 1/2
2. Substitute g(-2) into f(x):
f(x) = x^2 + 2x - 4
f(g(-2)) = f(1/2) (since g(-2) = 1/2)
= (1/2)^2 + 2(1/2) - 4
= 1/4 + 1 - 4
= 1/4 - 3¾
= 1/4 - 15/4
= -14/4
Therefore, f(g(-2)) = -14/4 = -7/2.