Domain and Range: What Goes In and What Comes Out

You can't take the square root of a negative number. You can't divide by zero. You can't take the logarithm of zero or a negative number.

These aren't arbitrary rules. They're boundaries.

Every function has boundaries. Some inputs are forbidden. Some outputs are impossible. The domain tells you which inputs work. The range tells you which outputs you can get.

Understanding domain and range means understanding where a function lives and what it can do.

The Unlock: Functions Have Territories

Think of a function as a machine. You feed in inputs, it spits out outputs.

But the machine has limits. Some things you try to feed in break the machine. Other things the machine is physically incapable of producing.

Domain: The set of inputs that don't break the machine.

Range: The set of outputs the machine can produce.

Domain is about what you're allowed to put in. Range is about what you can possibly get out.

Domain: What Inputs Are Valid?

For f(x) = x², the domain is all real numbers. You can square anything. 5² = 25. (-3)² = 9. π² ≈ 9.87. No input breaks the function.

For g(x) = √x, the domain is all non-negative numbers. You can take the square root of 0, 1, 100. But √(-1) isn't a real number. Negative inputs are forbidden.

For h(x) = 1/x, the domain is all non-zero numbers. You can divide by anything except zero. Division by zero is undefined. So zero is excluded from the domain.

The domain is determined by the operations in the function. Each operation imposes constraints.

Common Domain Restrictions

Square roots and even roots: The expression inside must be non-negative.

Example: f(x) = √(x - 3).

You need x - 3 ≥ 0, so x ≥ 3. Domain: [3, ∞).

Denominators: Cannot be zero.

Example: f(x) = 1/(x - 2).

You need x - 2 ≠ 0, so x ≠ 2. Domain: all real numbers except 2.

Logarithms: The argument must be positive.

Example: f(x) = ln(x).

You need x > 0. Domain: (0, ∞).

Multiple restrictions: Take the intersection.

Example: f(x) = √x / (x - 1).

Square root requires x ≥ 0. Denominator requires x ≠ 1. Domain: [0, 1) ∪ (1, ∞).

Notation for Domains

Domains are sets. We use interval notation.

• [a, b]: all numbers from a to b, including a and b.
• (a, b): all numbers from a to b, excluding a and b.
• [a, b): includes a, excludes b.
• (a, b]: excludes a, includes b.
• [a, ∞): all numbers ≥ a.
• (-∞, a): all numbers < a.

Example: Domain of f(x) = 1/x is (-∞, 0) ∪ (0, ∞). All numbers except zero.

The union symbol ∪ means "or." The domain is everything less than zero OR everything greater than zero.

Range: What Outputs Are Possible?

Range is trickier than domain. To find the range, you need to ask: What values can this function actually produce?

For f(x) = x², the range is [0, ∞). Squaring gives non-negative results. You can get 0 (when x = 0), 1 (when x = ±1), 100 (when x = ±10), any positive number. But you can't get -5. There's no real number that squares to -5.

For g(x) = 1/x, the range is all non-zero numbers. As x gets very large, 1/x gets close to zero but never equals zero. As x gets close to zero, 1/x blows up to infinity. You can get any positive or negative number, but not zero.

For h(x) = √x, the range is [0, ∞). Square roots of non-negative numbers are non-negative. You can't get a negative output.

Finding Range: Three Strategies

Strategy 1: Solve for x in terms of y.

Let y = f(x). Solve for x. The values of y that give valid x are the range.

Example: y = x² + 1.

Solve for x: x² = y - 1, so x = ±√(y - 1).

For x to be real, you need y - 1 ≥ 0, so y ≥ 1. Range: [1, ∞).

Strategy 2: Analyze the function's behavior.

Look at what happens as x increases, decreases, or approaches boundaries.

Example: f(x) = 1/x.

As x → ∞, f(x) → 0 (from above). As x → -∞, f(x) → 0 (from below). As x → 0⁺, f(x) → ∞. As x → 0⁻, f(x) → -∞.

The function gets arbitrarily close to zero but never equals it. It gets arbitrarily large (positive and negative). Range: all non-zero numbers.

Strategy 3: Graph it.

Sometimes the easiest way to see the range is to graph the function and observe which y-values the graph hits.

This isn't always rigorous, but it's often effective for understanding.

Range of Transformed Functions

If you know the range of a base function, you can deduce the range of a transformed version.

• If f has range [a, b], then f(x) + c has range [a + c, b + c]. Vertical shift.
• If f has range [a, b], then cf has range [ca, cb] (if c > 0) or [cb, ca] (if c < 0). Vertical stretch or flip.

Example: f(x) = x² has range [0, ∞).

g(x) = x² + 3 has range [3, ∞). Shifted up by 3.

h(x) = -x² has range (-∞, 0]. Flipped upside down.

Domain and Range for Piecewise Functions

A piecewise function has different rules on different parts of its domain.

Example: f(x) = x² if x < 0 f(x) = x + 1 if x ≥ 0

Domain: The union of the domains of each piece. Here, both pieces work for their specified ranges, so the domain is all real numbers.

Range: The union of the ranges of each piece.

For x < 0, f(x) = x² gives outputs in (0, ∞). (Squaring negative numbers gives positive results.)

For x ≥ 0, f(x) = x + 1 gives outputs in [1, ∞).

Range: (0, ∞) ∪ [1, ∞) = (0, ∞).

(The two intervals overlap, so the union is just (0, ∞).)

Domain and Range in Context

In applied problems, the domain and range are often restricted by context.

Example: A function models the height of a projectile.

h(t) = -16t² + 64t.

Mathematically, the domain is all real numbers. But physically, t represents time. Negative time doesn't make sense. The projectile hits the ground when h(t) = 0:

-16t² + 64t = 0 t(-16t + 64) = 0 t = 0 or t = 4.

The projectile is in the air from t = 0 to t = 4. Domain: [0, 4].

The range is the set of possible heights. The maximum height occurs at the vertex of the parabola.

t = -b/(2a) = -64/(2·-16) = 2.

h(2) = -16(4) + 64(2) = -64 + 128 = 64.

The height starts at 0, reaches 64, returns to 0. Range: [0, 64].

Context constrains the domain and range.

Why Domain Matters: Where Functions Break

Functions break when you ask them to do the impossible.

Dividing by zero is undefined. The expression 1/0 doesn't have a value. It doesn't equal infinity—it's simply not defined.

Taking the square root of a negative number gives a complex number, which is outside the real numbers. If you're working with real-valued functions, √(-1) doesn't exist.

Taking the logarithm of zero or a negative number is undefined (for real numbers). ln(0) would mean "e to what power gives 0?" There's no such power.

The domain tells you where the function works. Outside the domain, the function doesn't exist.

Why Range Matters: What Functions Can Achieve

The range tells you the function's reach. What values can it actually produce?

This matters in optimization problems. If you're maximizing or minimizing a function, the extreme values are in the range.

It matters in inverse problems. If you want to solve f(x) = b, there's a solution only if b is in the range of f.

It matters in modeling. If a function models a physical quantity, the range tells you the possible values of that quantity.

Domain and Range for Common Functions

Polynomial: f(x) = a_n x^n + ... + a_1 x + a_0.

Domain: all real numbers.

Range: depends on the degree. For even degree, range is [m, ∞) or (-∞, m] depending on the leading coefficient. For odd degree, range is all real numbers.

Rational function: f(x) = p(x)/q(x), where p and q are polynomials.

Domain: all real numbers except where q(x) = 0.

Range: usually all real numbers except a finite set of values (determined by horizontal asymptotes).

Square root: f(x) = √x.

Domain: [0, ∞).

Range: [0, ∞).

Exponential: f(x) = b^x (b > 0, b ≠ 1).

Domain: all real numbers.

Range: (0, ∞). Exponentials are always positive.

Logarithm: f(x) = log_b(x) (b > 0, b ≠ 1).

Domain: (0, ∞).

Range: all real numbers.

Trigonometric: sin and cos have domain all real numbers, range [-1, 1]. tan has domain all reals except odd multiples of π/2, range all real numbers.

Domain and Range for Compositions

If you compose two functions, the domain of the composition is more restrictive than either individual domain.

Let f(x) = √x and g(x) = x - 1.

The composition f(g(x)) = √(x - 1).

Domain of g: all real numbers.

Domain of f: [0, ∞).

Domain of f(g(x)): You need g(x) to be in the domain of f. So you need x - 1 ≥ 0, which means x ≥ 1.

Domain of f(g(x)): [1, ∞).

For the range, you trace through the composition.

g maps [1, ∞) to [0, ∞). Then f maps [0, ∞) to [0, ∞). So the range of f(g(x)) is [0, ∞).

Domain and Range for Inverses

If f has domain D and range R, then f⁻¹ has domain R and range D.

The inverse swaps inputs and outputs. So the domain of the inverse is the range of the original, and vice versa.

Example: f(x) = 2x has domain all real numbers and range all real numbers.

f⁻¹(x) = x/2 also has domain all real numbers and range all real numbers.

Example: f(x) = x² (restricted to x ≥ 0) has domain [0, ∞) and range [0, ∞).

f⁻¹(x) = √x has domain [0, ∞) and range [0, ∞).

The domain and range swap.

Restricted Domains

Sometimes you artificially restrict the domain to make a function behave better.

Example: f(x) = x² is not one-to-one. Both x = 2 and x = -2 give f(x) = 4. So there's no unique inverse.

But if you restrict to x ≥ 0, then f becomes one-to-one. Each output comes from exactly one input. Now you can define the inverse: f⁻¹(x) = √x.

Restricting the domain is common for trigonometric functions to define their inverses.

sin(x) oscillates between -1 and 1. Many different x values give the same output. But if you restrict to x ∈ [-π/2, π/2], sin becomes one-to-one. The inverse, arcsin(x), is defined on this restricted domain.

Implicit Domain

Sometimes the domain isn't stated explicitly. You have to figure it out from the function's formula.

This is called the implicit domain or natural domain: the largest set of real numbers for which the function is defined.

Example: f(x) = √(4 - x²).

You need 4 - x² ≥ 0, so x² ≤ 4, so -2 ≤ x ≤ 2.

Implicit domain: [-2, 2].

If a problem specifies a domain, use that. If not, assume the implicit domain.

Graphical Interpretation

On a graph, the domain is the set of x-values the graph covers. The range is the set of y-values.

Domain: Project the graph onto the x-axis. Which x-values have a corresponding point on the graph?

Range: Project the graph onto the y-axis. Which y-values have a corresponding point on the graph?

For a parabola y = x² - 1:

The graph extends left and right infinitely. Domain: all real numbers.

The lowest point is at y = -1, and the graph extends upward infinitely. Range: [-1, ∞).

Why Domain and Range Are Fundamental

Domain and range define the function's boundaries. They answer:

• Where does this function exist?
• What can this function produce?
• What are the function's constraints?

In calculus, you'll compute derivatives and integrals. But derivatives and integrals only make sense where the function is defined. The domain is the starting point.

In applied mathematics, domain and range encode physical constraints. Time is non-negative. Temperature has a lower bound. Probabilities are between 0 and 1.

Understanding domain and range means understanding the function's limits—both literally and conceptually.

Common Mistakes

Mistake 1: Confusing domain and range.

Domain is inputs. Range is outputs. They're different.

Mistake 2: Forgetting to check all restrictions.

If a function has a square root AND a denominator, both impose restrictions. The domain is where BOTH are satisfied.

Mistake 3: Assuming the range is all real numbers.

Many functions have restricted ranges. Always check.

Mistake 4: Ignoring context.

Mathematically, t can be any real number. Physically, t might represent time and must be non-negative.

Mistake 5: Treating domain as a formality.

Domain isn't just a box to check. It's where the function lives. Outside the domain, the function doesn't exist.

Domain and Range as Function Identity

Two functions are the same if and only if they have the same domain, the same range, and the same rule.

f(x) = x and g(x) = x²/x are not the same function.

For f, the domain is all real numbers.

For g, the domain is all non-zero numbers. (You can't have zero in the denominator.)

Even though x²/x simplifies to x (for x ≠ 0), g is not defined at x = 0. So g ≠ f.

Domain is part of the function's definition.

The Payoff: Seeing Structure

Once you understand domain and range, you start seeing structure.

You look at a function and immediately ask: Where is it defined? What can it produce? Where does it break?

You don't just compute values. You see boundaries.

This is the shift from procedural to conceptual thinking. You're not just following rules. You're understanding the landscape.

Domain and range are the first step in that shift.

Part 2 of the Precalculus series.

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