MathematicsRelations and FunctionsJEE

Relations and Functions · Part 10 of 20

Domain and Range of Algebraic Functions

Three tools for domain and range: completing the square, bounds for √ and 1/t, and the discriminant method, plus the six shapes of a quadratic by the signs of a and D.

Builds on: Part 9 · Inequalities and the Wavy Curve Method.

Video coming soon

Quadratic

Completing the square

$f (x) = a [(x + \frac{b}{2 a})^{2} - \frac{D}{4 a ^{2}}]$ , vertex $(- \frac{b}{2 a}, - \frac{D}{4 a})$ . x² − x − 3 = (x − ½)² − 13/4 has range [−13/4, ∞).

Parabola with vertex marked. — Range read from the vertex.

3 by 2 grid of parabolas. — Tinted: the always non-negative cases.

Six cases

Sign of a and D

a decides up or down; D < 0 no roots, D = 0 touches, D > 0 crosses. f(x) ≥ 0 for all x ⇔ a > 0 and D ≤ 0.

Discriminant method

(x² − x + 1)/(x² + x + 1)

(y − 1)x² + (y + 1)x + (y − 1) = 0 has a real root when (3 − y)(3y − 1) ≥ 0, and y = 1 works at x = 0. Both ends are reached: range [1/3, 3].

Graph with the range band. — Ends at (−1, 3) and (1, 1/3).

Summary

Key formulas

f (x) = a [(x + \frac{b}{2 a})^{2} - \frac{D}{4 a ^{2}}]

Quadratic function

a > 0 : range [- \frac{D}{4 a}, \infty)

Quadratic function

a < 0 : range (- \infty, - \frac{D}{4 a}]

Quadratic function

Worked examples

From the video

1. Range of √(x² − 4) and √(9 − x²).

[0, ∞) and [0, 3].

2. Domain and range of √(x² − 3x + 2) and √(x² − 4x + 6).

(−∞, 1] ∪ [2, ∞), [0, ∞); R, [√2, ∞).

3. Range of 1/(x² − x − 1).

(−∞, −4/5] ∪ (0, ∞).

4. All a such that the range of (x² − x)/(1 − ax) is R.

a > 1 (book prints [1, ∞), but a = 1 gives −x with x ≠ 1, missing −1).

JEE-style question

Your turn

The range of f(x) = (x² + x + 2)/(x² + x + 1), x ∈ R, is:

(a)(1, 7/3]

(b)[1, 7/3]

(c)(1, ∞)

(d)[7/3, ∞)

Show the answer and the traps

f = 1 + 1/t with t = x² + x + 1 ∈ [3/4, ∞). So 1/t ∈ (0, 4/3], and f ∈ (1, 7/3]. Option (a).

(b) includes 1, but 1/t is never 0. (c) forgets that t has a minimum.

(d) turns the bound the wrong way: 7/3 is the largest value, not the smallest.

Watch out

Common mistakes

Not checking the ends from D ≥ 0Test whether each end is reached; a = 1 above fails.

Forgetting the x² coefficient can vanishTreat y = 1 (or a = 0) separately.

Taking 1/t over an interval that contains 0 as one pieceSplit at t = 0.

Practice

Try these

1. Range of x² − 6x + 10.

[1, ∞).

2. Domain and range of √(x² − 6x + 8).

(−∞, 2] ∪ [4, ∞); [0, ∞).

3. Range of (x² + 1)/(x² + 2).

[1/2, 1).

4. Range of x/(x² + 1).

[−1/2, 1/2].

5. For which k is x² + kx + 4 > 0 for every real x?

−4 < k < 4.