Question

Let \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) such that \( g(x) \neq 0 \) for all \( x \in \mathbb{R} \), and \( f = f^{-1} \). Which of the following is correct?

• A. \( f \) must be discontinuous
• B. \( f \) is bijective and symmetric about \( y = x \)
• C. \( f \) is constant
• D. \( f \) is not differentiable anywhere

Hint:

Any function whose graph is symmetric about $y=x$ is its own inverse. A quick way to check is to see if $f(f(x)) = x$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A function \( f \) that satisfies \( f = f^{-1} \) is called an involution. For a function to have an inverse at all, it must be a bijection (both one-to-one and onto).

Step 2: Key Formula or Approach
If \( f(x) = y \), then \( f^{-1}(y) = x \). If \( f = f^{-1} \), then \( f(y) = x \). This means if the point \((x, y)\) lies on the graph, the point \((y, x)\) also lies on the graph.

Step 3: Detailed Explanation
1. Bijection: The existence of \( f^{-1} \) over the entire codomain \(\mathbb{R}\) implies that \( f \) is bijective. 2. Symmetry: The operation of swapping \( x \) and \( y \) coordinates corresponds to a reflection across the line \( y = x \). Since \( f(x) = y \implies f(y) = x \), the graph of the function is its own reflection. 3. Examples: Functions like \( f(x) = x \), \( f(x) = -x \), or \( f(x) = c - x \) are all examples of such functions. They are perfectly continuous and differentiable.

Step 4: Final Answer
The correct statement is that \( f \) is bijective and its graph is symmetric about the line \( y = x \).