If f(x)= x^2-4 and g(x)= 2x-7
Find
a) f(g inverse(x))
b) g((g)inverse(x))
To find the values of f(g^(-1)(x)) and g((g^(-1))(x)), we need to determine the inverse functions g^(-1)(x) and (g^(-1))(x) of g(x) first.
To find the inverse of g(x), we'll follow these steps:
Step 1: Replace g(x) with y: y = 2x - 7.
Step 2: Swap x and y: x = 2y - 7.
Step 3: Solve for y: x + 7 = 2y.
(x + 7)/2 = y.
Step 4: Replace y with g^(-1)(x): g^(-1)(x) = (x + 7)/2.
Now, let's proceed with the calculations:
a) f(g^(-1)(x)):
To calculate f(g^(-1)(x)), replace x in f(x) with g^(-1)(x):
f(g^(-1)(x)) = (g^(-1)(x))^2 - 4.
Substituting g^(-1)(x) = (x + 7)/2:
f(g^(-1)(x)) = ((x + 7)/2)^2 - 4.
Simplifying this expression gives the final result for a):
a) f(g^(-1)(x)) = (x^2 + 14x + 49)/4 - 4.
b) g((g^(-1))(x)):
Similarly, to calculate g((g^(-1))(x)), we substitute g^(-1)(x) into g(x):
g((g^(-1))(x)) = 2 * g^(-1)(x) - 7.
Substituting g^(-1)(x) = (x + 7)/2:
g((g^(-1))(x)) = 2 * ((x + 7)/2) - 7.
Simplifying this expression gives the final result for b):
b) g((g^(-1))(x)) = x + 7 - 7.
b) g((g^(-1))(x)) = x.
Therefore, the answers are:
a) f(g^(-1)(x)) = (x^2 + 14x + 49)/4 - 4.
b) g((g^(-1))(x)) = x.