Explain how to compose f^1(x)=x^2
and g^-1(x)=(1/-4)x+3
so that the composite function is equal to
h^-1(x)=3-x^2/4. Use your work to make a generalization about how to express the inverse of a composite function.
To find the inverse of a composite function, we will first find the inverses of the individual functions f(x) and g(x) and then compose them.
Given: f^1(x) = x^2 and g^-1(x) = (1/-4)x + 3
First, we will find the inverses of the given functions:
Inverse of f(x):
f(x) = x^2
To find f^1(x), we swap x and y:
x = y^2
Solving for y, we get:
y = sqrt(x)
Therefore, f^1(x) = sqrt(x)
Inverse of g(x):
g(x) = (1/-4)x + 3
To find g^-1(x), we swap x and y:
x = (1/-4)y + 3
Solving for y, we get:
y = (4/x - 12)/-1
Therefore, g^-1(x) = (4/x - 12)
Now, we compose f^1(x) and g^-1(x):
(f^1 o g^-1)(x) = f^1(g^-1(x))
= f^1((4/x - 12))
Substitute f^1(x) = sqrt(x) into:
= sqrt((4/x - 12))
= sqrt(4/x - 12)
= sqrt(4/x) - sqrt(12)
= 2/sqrt(x) - sqrt(12)
Therefore, (f^1 o g^-1)(x) = 2/sqrt(x) - sqrt(12) = h^-1(x) = 3 - x^2/4
From this, it can be generalized that to find the inverse of a composite function, first find the inverses of the individual functions and then compose them together.