Learn/Algebra/Domain and Range

Algebra • Topic 5

Domain and Range

Determining the domain and range of a function is often the first step in solving functional equations or analyzing inequalities. In olympiad problems, these sets can be restricted in subtle ways.

Definitions

Definition (Domain)

The domain of a function

f

is the set of all possible input values

x

for which the function is defined.

Definition (Range)

The range (or image) is the set of all possible output values

f (x)

x

varies over the domain.

Finding the Domain

To find the domain, look for "deal-breakers"—operations that are mathematically undefined.

Common Restrictions

Denominators: Cannot be zero.

$\frac{1}{P ( x )} ⟹ P (x) \neq = 0$ .

2. Even Roots: Radicand must be non-negative.

$2 n P (x) ⟹ P (x) \geq 0$ .

3. Logarithms: Argument must be strictly positive.

$lo g_{b} (P (x)) ⟹ P (x) > 0$ .

Example

Find the domain of

f (x) = x^{2} - 4 + \frac{1}{l n ( x )}

Proof (Solution). We have three constraints:

Square root: $x^{2} - 4 \geq 0 ⟹ x \in (- \infty, - 2] \cup [2, \infty)$ .
Logarithm argument: $x > 0$ .
Denominator: $ln (x) \neq = 0 ⟹ x \neq = 1$ .

Intersecting these sets:

$(- \infty, - 2]$ is ruled out by $x > 0$ .
$[2, \infty)$ satisfies $x > 0$ and $x \neq = 1$ .

Domain:

[2, \infty)

. ∎

Finding the Range

Finding the range is often harder. Common techniques include:

1. Discriminant Method

y = \frac{a x ^{2} + b x + c}{d x ^{2} + e x + f}

, rewrite as a quadratic in

x

with coefficients involving

y

. For real

x

to exist, the discriminant

Δ_{x} \geq 0

. This gives an inequality for

y

Example

Find the range of

y = \frac{x ^{2}}{x ^{2} + 1}

Proof. $y (x^{2} + 1) = x^{2} ⟹ y x^{2} + y = x^{2} ⟹ (y - 1) x^{2} + y = 0$ .

If $y = 1$ , we get $1 = 0$ (impossible), so $y \neq = 1$ .
For quadratic in $x$ , we need $x^{2} = \frac{- y}{y - 1} = \frac{y}{1 - y} \geq 0$ .

Solving $\frac{y}{1 - y} \geq 0$ yields $y \in [0, 1)$ .

Range:

[0, 1)

. ∎

2. AM-GM and Inequalities

Use standard inequalities to bound the function.

$x + \frac{1}{x} \geq 2$ for $x > 0$ .
$- 1 \leq sin x \leq 1$ .

Practice Problems

Exercise (Problem 1)

Find the domain of

f (x) = \frac{x - 1}{x + 2}

Exercise (Problem 2)

Find the range of

f (x) = 3 sin (x) + 4 cos (x)

Exercise (Problem 3)

Find the range of

y = \frac{x + 2}{x ^{2} + 3}