Sets, Relations & Functions Question - None

Question

Let $f: \mathbb{R} \to \mathbb{R}$ , $f(x) = \dfrac{2x^2-5x+3}{8x^2+9x+11}$ . Determine the nature of $$f$$ .

Accepted Answer

Key Idea: Test onto via discriminant, test one-one via derivative sign changes.

Step 1: Denominator is never zero: $\Delta = 81 - 352 < 0$ . Domain is all $\mathbb{R}$ .

Step 2 (Onto): Set $$y = f(x)$$ , rearrange to quadratic in $$x$$ . Discriminant condition gives $1 + 226y - 271y^2 \ge 0$ — not true for all real $$y$$ . So not onto.

Step 3 (One-one): $$f'(x)$$ has real roots, so $$f'$$ changes sign. So not one-one.

Answer: Many-one into

Trap: Having domain

\mathbb{R}

does not imply range

\mathbb{R}

. Equal-degree rational functions often miss an interval.

Key Concept: Range of rational function via discriminant; injectivity via derivative

Question:

Step-by-Step Solution

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