Question

The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = e^x + e^{-x} \) is:

• A. One-one
• B. Onto
• C. Bijective
• D. Not bijective

Hint:

For exponential functions, analyze monotonicity using derivatives and determine the range for ”onto” checks.

Answer 1

The Correct Option is D

1. The function \(f(x) = e^x + e^{-x}\) is defined for all \(x \in \mathbb{R}\).

2. To determine if f is injective (one-to-one): - Calculate the derivative:

\[f'(x) = e^x - e^{-x}.\]

- As \(f'(x) > 0\) for \(x > 0\) and \(f'(x) < 0\) for \(x < 0\), \(f(x)\) increases strictly for \(x > 0\) and decreases strictly for \(x < 0\). Consequently, \(f(x)\) is not injective.

3. To determine if f is surjective (onto): - The range of \(f(x)\) is:

\[f(x) = e^x + e^{-x} \ge 2 \quad \text{for all } x \in \mathbb{R}.\]

- Because \(f(x)\) does not cover all real numbers (\(f(x) \ge 2\)), \(f(x)\) is not surjective.

4. Since \(f(x)\) is neither injective nor surjective, it is not bijective.