Question 19 of 33intermediate🔧 ApplyShort Answer4 marks

Find the range of each of the following functions.

(i)

f(x)=2−3x,x∈R,x>0f(x) = 2 - 3x, x \in R, x > 0.

Answer

Range = (-∞, 2) or {y : y ∈ R, y < 2}

(ii)

f(x)=x2+2,x is a real numberf(x) = x^2 + 2, x \text{ is a real number}.

Answer

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

(iii)

f(x)=x,x is a real numberf(x) = x, x \text{ is a real number}.

Answer

Range = R (all real numbers)

Explanation

For part (i), since x > 0, as x approaches 0 from the right, f(x) approaches 2, and as x increases indefinitely, f(x) decreases indefinitely. Since x cannot equal 0, f(x) never reaches 2. For part (ii), x² ≥\geq 0 for all real x, so the minimum value is 2 when x = 0. For part (iii), the identity function maps every real number to itself, so the range equals the domain.

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