Question

F(x) = (x + 3)/(x-2) and g(x) = (2x + 3)/(x-1). confirm that they are inverses. ​

Answer 1

To confirm that f(x) = (x+3)/(x-2) and g(x) = (2x+3)/(x-1) are inverses, we compose them as f(g(x)) and g(f(x)). If both compositions return the identity function, which means the result is simply x, the original input, then the functions are inverses.

Step-by-step explanation:

To confirm if two functions are inverses of each other, one should compose one function with the other in both orders and see if the result is the identity function, which means the output is equal to the input.

Let's start with f(g(x)) and see if it yields x:

1. f(g(x)) = f((2x + 3)/(x-1))
2. Now substitute g(x) into f(x): f(x) = (x + 3)/(x-2), we get:
3. f(g(x)) = ((2x + 3)/(x-1) + 3) / ((2x + 3)/(x-1) - 2)
4. After simplifying, this should give us the original x back if they are indeed inverses.

Next, we check g(f(x)):

1. g(f(x)) = g((x + 3)/(x-2))
2. Substituting f(x) into g(x): g(x) = (2x + 3)/(x-1), we get:
3. g(f(x)) = (2(x + 3)/(x-2) + 3) / ((x + 3)/(x-2) - 1)
4. Similarly, after simplifying, this should also give us x if f(x) and g(x) are inverses.

If both compositions yield the identity function, then we can confirm that f(x) and g(x) are indeed inverses of each other.