F(x) = 1-x and g(x) = 1/x
These functions have the property that f = f^-1(inverse) and g = g ^-1. That is, the inverse of f is equal to itself and the inverse of g is also equal to itself. Take the composition of each function with itself to show that this is true. Remember, if the function h(x) is equal to its own inverse, then h o h^-1 = x.
To show that f(x) = 1 - x is equal to its own inverse, we need to find the inverse function f^(-1)(x) and verify that f(f^(-1)(x)) = x.
Step 1: Find the inverse function of f(x)
To find the inverse function, we swap the variables x and y and solve for y.
So, let's write f(x) = y and interchange x and y:
x = 1 - y
Solve for y:
y = 1 - x
Therefore, the inverse function of f(x) = 1 - x is f^(-1)(x) = 1 - x.
Step 2: Verify that f(f^(-1)(x)) = x
Take the composition of f with its inverse function and check if it equals x.
f(f^(-1)(x)) = f(1 - x)
Substitute y = 1 - x in f(x):
f(1 - x) = 1 - (1 - x) = 1 - 1 + x = x
Thus, f(f^(-1)(x)) = x, which means that f(x) = 1 - x is equal to its own inverse.
Similarly, let's verify that g(x) = 1/x is equal to its own inverse by finding the composition g(g^(-1)(x)) = x.
Step 1: Find the inverse function of g(x)
To find the inverse function, we swap the variables x and y and solve for y.
So, let's write g(x) = y and interchange x and y:
x = 1/y
Solve for y:
y = 1/x
Therefore, the inverse function of g(x) = 1/x is g^(-1)(x) = 1/x.
Step 2: Verify that g(g^(-1)(x)) = x
Take the composition of g with its inverse function and check if it equals x.
g(g^(-1)(x)) = g(1/x)
Substitute y = 1/x in g(x):
g(1/x) = 1/(1/x) = x
Hence, g(g^(-1)(x)) = x, which means g(x) = 1/x is equal to its own inverse.