Problem
The keyword “domain and range of continuous graphs” asks for a reliable method to determine the domain and range from a graph that is continuous (drawn without breaks on the interval shown).
Definitions
Domain: the set of all real \(x\)-values for which the graph has at least one point \((x,y)\).
Range: the set of all real \(y\)-values that occur on the graph for some \(x\).
How to read domain and range from a continuous graph
1) Domain: project the graph onto the \(x\)-axis
Identify the leftmost \(x\)-value and rightmost \(x\)-value reached by the graph. For a continuous graph segment with endpoints, the domain is the interval between those \(x\)-values.
Endpoint rule: a filled endpoint means the endpoint \(x\)-value is included (use a bracket). An open circle means it is excluded (use a parenthesis). An arrow means the graph continues without bound (use \( \pm\infty \)).
2) Range: project the graph onto the \(y\)-axis
Identify the lowest \(y\)-value and highest \(y\)-value reached by the graph. For a continuous graph on an interval, the range is the interval between those \(y\)-values, with inclusion determined by whether the extreme values are actually attained on the graph.
Worked example (a typical continuous graph)
Consider a continuous parabola-shaped graph segment whose endpoints are included, with a highest point (maximum) inside the interval. The curve corresponds to the function
\[ y = -(x-1)^2 + 5 \quad \text{for } -2 \le x \le 4. \]
Step A: Domain
The graph starts at \(x=-2\) (filled endpoint) and ends at \(x=4\) (filled endpoint). Therefore,
\[ \text{Domain} = [-2,4]. \]
Step B: Range
The maximum occurs at the vertex \(x=1\), giving
\[ y_{\max} = -(1-1)^2 + 5 = 5. \]
The minimum occurs at the endpoints (symmetry places both endpoints at the same height):
\[ y(-2)= -(-2-1)^2+5 = -9+5=-4,\qquad y(4)= -(4-1)^2+5 = -9+5=-4. \]
Therefore,
\[ \text{Range} = [-4,5]. \]
Visualization
Summary table
| Quantity | How it is read from a continuous graph | Example value |
|---|---|---|
| Domain | Leftmost to rightmost \(x\)-values, with brackets/parentheses from endpoint style | \( [-2,4] \) |
| Range | Lowest to highest \(y\)-values, with brackets/parentheses from whether extremes occur on the graph | \( [-4,5] \) |
| Continuity benefit | A continuous graph on an interval produces a connected range (an interval) | \( [-4,5] \) is an interval |
Common pitfalls
Do not confuse domain and range: domain is about \(x\)-values (horizontal), range is about \(y\)-values (vertical).
Endpoints matter: filled endpoints use brackets; open circles use parentheses.
Arrows imply infinity: if a continuous graph extends left or right, domain may include \( -\infty \) or \( +\infty \); similarly for range if the graph extends upward or downward without bound.
Final answer
For domain and range of continuous graphs, the domain is the full set of \(x\)-values covered by the graph and the range is the full set of \(y\)-values covered; in the example shown, the domain is \( [-2,4] \) and the range is \( [-4,5] \).