Finding the domain and range of a curved line involves identifying the set of all possible x-values (domain) and y-values (range) that the curve covers. The reference states that the set of all x-coordinates of all points of the curve would give the domain and the set of all y-coordinates of all points of the curve would give the range.
Understanding Domain and Range
- Domain: All possible input values (x-values) for which the curve is defined.
- Range: All possible output values (y-values) that the curve produces.
Both domain and range can be expressed as a set or an interval.
Steps to Determine Domain and Range
Here's a breakdown of how to find the domain and range, especially when dealing with a curved line:
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Visualize the Curve: If you have a graph, visually inspect the curve. If you have an equation, try to sketch a rough graph or use graphing software.
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Determine the Domain:
- Look for Horizontal Extent: How far left and right does the curve extend on the x-axis? This determines the minimum and maximum x-values.
- Identify Restrictions: Are there any x-values where the curve is undefined? This could be due to:
- Vertical Asymptotes: The curve approaches but never touches a vertical line. The x-value of this line is excluded from the domain.
- Discontinuities/Holes: A point where the curve is not defined.
- Square Roots/Logarithms: If your equation has square roots, the expression inside must be greater than or equal to zero. If it has a logarithm, the expression inside must be greater than zero.
- Express as Interval/Set: Write the domain as an interval (e.g.,
(-∞, ∞),[0, 5],(-2, 3) U (3, ∞)) or a set of x-values.
-
Determine the Range:
- Look for Vertical Extent: How far down and up does the curve extend on the y-axis? This determines the minimum and maximum y-values.
- Identify Restrictions: Are there any y-values that the curve never reaches? This could be due to:
- Horizontal Asymptotes: The curve approaches but never touches a horizontal line. The y-value of this line may be excluded from the range.
- Maximum/Minimum Points: The curve might have a highest or lowest point, limiting the range.
- Express as Interval/Set: Write the range as an interval (e.g.,
(-∞, ∞),[0, 5],(-2, 3) U (3, ∞)) or a set of y-values.
Examples
Here are some examples to illustrate the process:
| Curve | Domain | Range | Notes |
|---|---|---|---|
| Straight line y = x | (-∞, ∞) |
(-∞, ∞) |
Extends infinitely in both x and y directions |
| Parabola y = x2 | (-∞, ∞) |
[0, ∞) |
All real numbers for x, but y is always non-negative |
| y = 1/x | (-∞, 0) U (0, ∞) |
(-∞, 0) U (0, ∞) |
Excludes x=0 due to vertical asymptote, y=0 for horizontal |
| Circle x2 + y2 = r2 | [-r, r] |
[-r, r] |
Circle centered at origin with radius r |
| Curve y = √x | [0, ∞) |
[0, ∞) |
x cannot be negative, y is always non-negative |
Practical Insights
- Graphing Tools: Utilize graphing calculators or online tools (like Desmos or Wolfram Alpha) to visualize the curve and help determine its domain and range.
- Piecewise Functions: If the curve is defined by different equations over different intervals (a piecewise function), analyze each piece separately.
- Asymptotes are Important: Pay close attention to asymptotes as they often define the boundaries of the domain and range.
