Find the range of each of the following functions.

Question 23 of 35intermediate🔧 ApplyShort Answer6 marks

(i)

$f(x) = 2 - 3x$ , $x \in \mathbb{R}$ , $x > 0$ .

Answer

Range = (-∞, 2) or {y : y ∈ ℝ, y < 2}

(ii)

$f(x) = x^2 + 2$ , $x$ is a real number.

Answer

Range = [2, ∞) or {y : y ∈ ℝ, y $\geq$ 2}

(iii)

$f(x) = x$ , $x$ is a real number.

Answer

Range = ℝ (all real numbers)

Explanation

For part (i), since x > 0, the function f(x) = 2 - 3x is strictly decreasing. As x → 0⁺, f(x) → 2, and as x → ∞, f(x) → -∞. Thus range is (-∞, 2). For part (ii), since x² $\geq$ 0 for all real x, the minimum value of f(x) = x² + 2 is 2 (at x = 0), and it can take any value $\geq$ 2. For part (iii), the identity function f(x) = x maps every real number to itself, so the range equals the domain, which is ℝ.

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