AP DSC School Assistant Math Model Test 3

Concept:A function is one‑one (injective) if different inputs always give different outputs, i.e., f(x₁) = f(x₂) ⇒ x₁ = x₂. It is onto (surjective) if every real number y has some x such that f(x) = y.Explanation:Check one‑one: Let f(x₁) = f(x₂).$\frac{x_1}{1+x_1^2} = \frac{x_2}{1+x_2^2}$Cross‑multiply: $x_1(1+x_2^2) = x_2(1+x_1^2)$⇒ $x_1 + x_1x_2^2 = x_2 + x_2x_1^2$⇒ $x_1 - x_2 = x_1x_2(x_1 - x_2)$⇒ $(x_1 - x_2)(1 - x_1x_2) = 0$So either $x_1 = x_2$ or $x_1x_2 = 1$.Hence, for example, f(2) = f(1/2) = 0.4 but 2 ≠ 1/2. Thus f is not one‑one.Check onto: For a real y, solve $\frac{x}{1+x^2} = y$.⇒ $x = y(1+x^2) = y + yx^2$⇒ $yx^2 - x + y = 0$.This quadratic has real x only if discriminant ≥ 0:$1 - 4y^2 ≥ 0$ ⇒ $|y| ≤ \frac{1}{2}$.Thus for y = 2 (or any |y| > 0.5), no real x exists. So f is not onto (its range is $[-\frac12,\frac12]$).Answer:The function is neither one‑one nor onto. Hence the correct option is B.