Question

If \( f(x) = \frac{2x + 3}{3x - 2}, \, x \neq \frac{2}{3}, \) then the function \( f \circ f \) is

• A. an even function
• B. an identity function
• C. a constant function
• D. an exponential function

Hint:

When finding compositions of functions, always substitute the function into itself and simplify the expression to determine the result.

Answer 1

The Correct Option is B

Step 1: Understanding the composition of functions.
We are tasked with finding \( f \circ f \), which means applying the function \( f(x) \) to itself. This requires substituting \( f(x) \) into itself. The function \( f(x) \) is given by: \[ f(x) = \frac{2x + 3}{3x - 2} \]
Step 2: Substituting \( f(x) \) into itself.
We compute \( f(f(x)) \) by substituting \( f(x) \) into the formula for \( f(x) \): \[ f(f(x)) = f\left( \frac{2x + 3}{3x - 2} \right) \] This results in an identity function because applying \( f \) to itself yields the original input.

Step 3: Conclusion.
Therefore, \( f \circ f \) is an identity function, which makes option (B) the correct answer.