Question

Find the range of each of the following functions.
(i) f(x) = 2 - 3x, x ∈ R, x> 0.
(ii) f(x) = x²+ 2, x, is a real number.
(iii) f(x) = x, x is a real number

Answer 1

(i) Find the range of f(x) = 2 − 3x, x ∈ R, x > 0

Given x > 0.

Since f(x) = 2 − 3x is a linear function with negative slope, as x increases, f(x) decreases.

When x → 0⁺,
f(x) → 2

When x → ∞,
f(x) → −∞

So, the range is:

(−∞, 2)

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(ii) Find the range of f(x) = x² + 2, x ∈ R

Since x² ≥ 0 for all real x,

x² + 2 ≥ 2

The minimum value of f(x) occurs at x = 0:

f(0) = 2

As |x| → ∞, f(x) → ∞

So, the range is:

[2, ∞)

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(iii) Find the range of f(x) = x, x ∈ R

Here, f(x) = x can take every real value.

So, the range is:

(−∞, ∞)

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Final Answers:

(i) Range = (−∞, 2)
(ii) Range = [2, ∞)
(iii) Range = (−∞, ∞)