Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^7 + 5x^3 + 11x + 15 \). Then, which of the following statements is TRUE?

• A. \( f \) is both one-one and onto
• B. \( f \) is neither one-one nor onto
• C. \( f \) is one-one but NOT onto
• D. \( f \) is onto but NOT one-one

Hint:

For a polynomial of odd degree with a positive leading coefficient and a positive derivative everywhere, the function is strictly increasing and hence both one-one and onto.

Answer 1

The Correct Option is A

Step 1: Analyze the function.
The given function is \( f(x) = x^7 + 5x^3 + 11x + 15 \), which is a polynomial of odd degree (7).
Step 2: Check monotonicity.
Derivative: \[ f'(x) = 7x^6 + 15x^2 + 11 \] Since \( x^6, x^2 \ge 0 \), \( f'(x)>0 \) for all \( x \in \mathbb{R} \). Thus, \( f(x) \) is a strictly increasing function.
Step 3: Check one-one and onto nature.
Because \( f(x) \) is strictly increasing, it is one-one (injective). As the degree is odd, the limits are: \[ \lim_{x \to \infty} f(x) = \infty, \quad \lim_{x \to -\infty} f(x) = -\infty \] Hence, the range covers all real numbers \( \mathbb{R} \), making it onto (surjective). Final Answer: \[ \boxed{f \text{ is both one-one and onto.}} \]