F(x) = 1-x and g(x) = 1/x
Write in the form (ax + b)/(cx + d)
a.What are all the possible values of a, b, c, and d?
b.Find all the functions fo the form f (x) = (ax + b)/(cx + d) with the property that
f o f^-1 = x
To write the functions f(x) = 1-x and g(x) = 1/x in the form (ax + b)/(cx + d), we can follow these steps:
a. For f(x) = 1-x:
- Let's start by assuming the given form: (ax + b)/(cx + d)
- Compare the given form with f(x) = 1-x.
- We can see that a = -1, b = 1, c = -1, and d = 1.
- Therefore, the possible values of a, b, c, and d for f(x) = 1-x are: a = -1, b = 1, c = -1, d = 1.
b. For g(x) = 1/x:
- Similarly, let's start with the form (ax + b)/(cx + d).
- Compare the given form with g(x) = 1/x.
- We can see that a = 1, b = 0, c = 1, and d = 1.
- Therefore, the possible values of a, b, c, and d for g(x) = 1/x are: a = 1, b = 0, c = 1, d = 1.
Now, let's focus on the second part of the question:
b. To find the functions f(x) = (ax + b)/(cx + d) with the property f o f^(-1) = x:
- First, we need to find the inverse function of f(x).
- For f(x) = 1-x, we can find its inverse as follows:
- Let y = f(x) = 1-x
- Swap x and y and solve for y: x = 1-y
- Rearrange the equation to solve for y: y = 1-x
- Therefore, the inverse of f(x) is f^(-1)(x) = 1-x.
- Now, let's substitute f(x) and f^(-1)(x) into the equation f o f^(-1) = x and simplify:
- (f o f^(-1))(x) = f(f^(-1)(x)) = f(1-x)
- = 1 - (1-x) = 1 - 1 + x = x
Since this simplifies to x, we can see that any function of the form f(x) = (ax + b)/(cx + d) where a = -1, b = 1, c = -1, and d = 1 will satisfy the property f o f^(-1) = x.