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:
Given that \[ f = f^{-1} \] this means the function is equal to its own inverse. Such a function is called an involution.
Step 2: Key Property:
A function has an inverse only if it is bijective, i.e.
one-to-one (injective)
onto (surjective)
Hence, $f$ must be bijective.
Step 3: Graphical Interpretation:
The graph of a function and its inverse are always mirror images of each other about the line \[ y=x \] Since \[ f = f^{-1} \] the graph remains unchanged after reflection in the line $y=x$. Therefore, the graph of $f$ is symmetric about the line $y=x$.
Step 4: Checking Other Options:
[(A)] False, because $f(x)=-x$ is continuous and satisfies $f=f^{-1}$.
[(C)] False, because a constant function is not one-to-one, so inverse does not exist.
[(D)] False, because $f(x)=x$ is differentiable everywhere and satisfies $f=f^{-1}$.
Step 5: Final Answer: \[ \boxed{(B)\ \text{$f$ is bijective and symmetric about } y=x} \]