Question

If f(x) = x²/(2x - 1) and g(x) = 2x - 3, find expressions in x for (f ∘ g)(x) and (g ∘ f)(x).

Answer 1

The expression for (f ∘ g)(x) is (4x² - 12x + 9)/(4x - 7), and the expression for (g ∘ f)(x) is (2x² - 6x + 3)/(2x - 1) after performing function composition.

Step-by-step explanation:

To find the expressions for (f ∘ g)(x) and (g ∘ f)(x), we need to perform function composition, which involves substituting one function into another.

For (f ∘ g)(x), we substitute g(x) into f(x):

1. First, find g(x):
   g(x) = 2x - 3.
2. Then, substitute g(x) into f(x):
   f(g(x)) = f(2x - 3) = (2x - 3)²/[2(2x - 3) - 1].
3. Simplify the expression:
   f(g(x)) = (2x - 3)²/(4x - 6 - 1) = (4x² - 12x + 9)/(4x - 7).

Next, for (g ∘ f)(x), we substitute f(x) into g(x):

1. First, find f(x):
   f(x) = x²/(2x - 1).
2. Then, substitute f(x) into g(x):
   g(f(x)) = g(x²/(2x - 1)) = 2(x²/(2x - 1)) - 3.
3. Simplify the expression:
   g(f(x)) = (2x²)/(2x - 1) - 3(2x - 1)/(2x - 1) = (2x² - 6x + 3)/(2x - 1).

So, the expressions for function compositions are (f ∘ g)(x) = (4x² - 12x + 9)/(4x - 7) and (g ∘ f)(x) = (2x² - 6x + 3)/(2x - 1).