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Mathematics › 🌆 Grade 9 › Shadow Projector: Domain and Range
🌆 Grade 9 · Lesson 2 of 12

Shadow Projector: Domain and Range

Shine a light straight down on a curve and its shadow on the x-axis is the domain; shine it sideways and the shadow on the y-axis is the range.

Grade 9Algebra 1
Shadow Projector: Domain and Range — illustration
💡
The big idea: Every function has two sets attached to it: the inputs it is willing to accept (the domain) and the outputs it can actually produce (the range). If you project a graph straight down onto the x-axis, the “shadow” it casts is exactly the domain; project it sideways onto the y-axis and the shadow is the range.
🎯 By the end, you'll be able to
  • Define the domain and range of a function
  • Read the domain and range of a function from its graph
  • Determine the domain of a simple algebraic function by finding what values of x are not allowed
  • Express domain and range using inequality notation
📎 You should already know
  • Function notation (f(x))
  • The coordinate plane

Casting a shadow on each axis

Picture a curve drawn in the air above a coordinate grid, lit from directly above. Its shadow falls straight down onto the x-axis, covering exactly the x-values where the curve exists. Now imagine a light shining sideways instead: the shadow falls onto the y-axis, covering exactly the heights the curve reaches.

Those two shadows have names. The shadow on the x-axis is the domain — every input the function is allowed to take. The shadow on the y-axis is the range — every output the function actually produces.

🔑 Domain = inputs, Range = outputs
The domain of a function is the complete set of x-values (inputs) for which it is defined. The range is the complete set of y-values (outputs) it produces. Domain lives on the x-axis; range lives on the y-axis.
\[ \text{Domain: } -3 \le x \le 5 \qquad \text{Range: } 0 \le y \le 4 \]
A sample domain and range written with inequalities — the graph's x-shadow spans −3 to 5, and its y-shadow spans 0 to 4.
🎮 Shadow Projector LIVE
Project a curve onto the axes: its shadow on x is the domain, its shadow on y is the range.

Reading domain and range from a graph

To read the domain, scan the graph from left to right and note every x-value the curve actually passes over. To read the range, scan from bottom to top and note every y-value the curve actually reaches. Gaps, endpoints, and arrows that continue forever all matter.

📝 Worked example: Find the domain and range of f(x) = √(x − 2).
  1. A square root needs a non-negative number inside it, so we need x − 2 ≥ 0.
  2. Solving gives x ≥ 2 — that is the domain.
  3. A square root itself is never negative, and it grows without bound as x grows, so the outputs cover every value from 0 upward — that is the range.
✓ Domain: <strong>x &ge; 2</strong>. Range: <strong>y &ge; 0</strong>.
📝 Worked example: Find the domain of f(x) = 1 &divide; (x &minus; 3).
  1. Division by zero is undefined, so the denominator x − 3 cannot equal 0.
  2. That means x cannot equal 3.
  3. Every other real number is a valid input.
✓ Domain: <strong>all real numbers except x = 3</strong>.
✨ Restrictions come from the operations used
Domain restrictions almost always trace back to two operations: you cannot divide by zero, and you cannot take the square root of a negative number (in the real numbers). Scan an expression for a denominator or a square root, and you have found where to check.
⚠️ Don't mix up which axis is which
It is easy to swap domain and range by accident. Anchor it with the shadow picture: domain falls on the x-axis (inputs, read left-to-right), range falls on the y-axis (outputs, read bottom-to-top).

Domain and range describe real situations

If f(t) gives a rocket's height t seconds after launch, the domain is every valid time (probably t ≥ 0, until it lands), and the range is every height it actually reaches (from ground level up to its peak). Reading domain and range is really reading the boundaries of a real situation.

Check your understanding

1. A graph's curve exists only for x-values from &minus;2 to 6, inclusive. What is its domain?
The domain is the full shadow on the x-axis, from the smallest to the largest x the curve reaches: −2 ≤ x ≤ 6.
2. What is the domain of f(x) = &radic;(x &minus; 4)?
The expression under the square root must be ≥ 0, so x − 4 ≥ 0, which gives x ≥ 4.
3. What value must be excluded from the domain of f(x) = 1 &divide; (x + 5)?
The denominator x + 5 cannot equal 0, so x cannot equal −5.
4. The domain of a function is the set of…
Domain is the complete set of valid inputs (x-values) for the function.
5. A downward-opening parabola has its highest point at (0, 5) and continues forever in both horizontal directions. What is its range?
Since the parabola opens downward from a peak of 5, every output is 5 or less: y ≤ 5. That is the shadow on the y-axis.
✅ Key takeaways
  • Domain is the complete set of valid inputs (x-values); range is the complete set of outputs (y-values) the function produces.
  • Picture domain as the graph's shadow on the x-axis, and range as its shadow on the y-axis.
  • Common domain restrictions come from division by zero and square roots of negative numbers.
  • Read domain left-to-right along the graph and range bottom-to-top.
  • Domain and range describe the real boundaries of the situation a function models.