1. Rational Functions: Spotting Asymptotes & Discontinuities
A rational function is formed by dividing two polynomials: f(x) = P(x) / Q(x). When looking at its graph:
Vertical Asymptotes (Domain Gaps)
A vertical dotted line where the graph explodes toward +∞ or -∞. The graph will never touch or cross a vertical asymptote.
Domain: (-∞, 3) ∪ (3, ∞)
Horizontal Asymptotes (Range Boundaries)
A horizontal line that the graph approaches as x heads toward +∞ or -∞. Tells you what y-values are unreachable in the long run.
Range: (-∞, 0) ∪ (0, ∞)
2. Square Root & Radical Functions
Graphs of square root functions look like half-parabolas tipped on their side. They start at a single solid endpoint (vertex) and curve off in one direction:
Worked ExampleAnalyzing f(x) = √(x - 4) + 2
1. Starting Point on Graph: The curve begins at solid point (4, 2) and bends upward and rightward.
2. Domain Condition: Radicand x - 4 ≥ 0 &implies; x ≥ 4.
→ Domain: [4, ∞)
3. Range Condition: Since √(x - 4) ≥ 0, adding 2 gives y ≥ 2.
→ Range: [2, ∞)
3. Master Reference: Parent Function Domains & Ranges
Bookmark this reference table of standard parent functions:
| Parent Function | Graph Shape | Domain | Range |
|---|---|---|---|
| f(x) = x | Straight diagonal line | (-∞, ∞) | (-∞, ∞) |
| f(x) = x2 | Parabola opening up | (-∞, ∞) | [0, ∞) |
| f(x) = |x| | V-shaped graph | (-∞, ∞) | [0, ∞) |
| f(x) = √x | Half-parabola curve | [0, ∞) | [0, ∞) |
| f(x) = 1/x | Hyperbola with asymptotes | (-∞, 0) ∪ (0, ∞) | (-∞, 0) ∪ (0, ∞) |
| f(x) = 2x | Exponential curve | (-∞, ∞) | (0, ∞) |
4. Handling Piecewise Function Graphs
Piecewise graphs stitch different equations across different sections of the domain. When reading domain and range on a piecewise graph:
- Check where the pieces meet: If one piece ends with a solid dot (•) and the other begins with an open circle (○) at the same x-value, that x-value IS included in the domain!
- Watch for empty jumps: If there is a horizontal gap between two pieces where no graph exists, use a union symbol
∪to skip the gap:[-5, -1) ∪ [2, 7]. - Trace overlapping vertical ranges: Even if there is a jump discontinuity, the outputs of the two pieces might overlap in y-values, creating a continuous single range interval.
5. Frequently Asked Questions (FAQ)
How do you find the domain of a rational function with an asymptote?
Look for the vertical asymptote where the curve splits and shoots off toward positive or negative infinity. If a vertical asymptote is at x = c, x cannot equal c. In interval notation, the domain is written as (-∞, c) ∪ (c, ∞).
Why does a square root function have a restricted domain and range?
In real numbers, you cannot take the square root of a negative value. Therefore, the expression inside the radical must be greater than or equal to zero. Similarly, principal square roots produce non-negative outputs, restricting the range to values above the curve's starting vertex.
What is the difference between an asymptote and a hole in a graph?
An asymptote occurs when a factor in the denominator cannot be cancelled, causing the function to explode to infinity as x approaches that value. A hole (removable discontinuity) occurs when a common factor cancels from both numerator and denominator, leaving an isolated open circle at that single point.
What is the domain and range of an exponential function like f(x) = 2^x?
The domain is all real numbers (-∞, ∞) because you can raise 2 to any positive, negative, or zero power. The range is (0, ∞) because positive bases raised to any real power always produce strictly positive outputs, bounded below by a horizontal asymptote at y = 0.
Can you verify domain and range algebraically after looking at a graph?
Yes. Always check the two primary algebraic red flags: set any denominators equal to zero to find vertical asymptotes (domain exclusions), and set any expressions inside even radicals greater than or equal to zero.