3.1 Function Basics
Key Takeaways
- A function assigns exactly one output to each input; the vertical line test confirms this on a graph.
- Function notation f(x) names the output rule; f(3) means substitute 3 for x and evaluate.
- Domain is the set of allowed inputs (x-values); range is the set of resulting outputs (y-values).
- Key graph features Regents tests in context: intercepts, intervals of increase/decrease, maximum/minimum, and end behavior.
- On the Algebra I Regents, interpreting what a feature means in a real situation earns as much credit as computing it.
Why Functions Matter on the Regents
The Functions strand is 24%-32% of Algebra I Regents credits, second only to Algebra. Nearly every part of the exam uses function notation, so a shaky grasp here costs points across multiple-choice and constructed-response tasks.
The Regents rewards interpretation, not just calculation. A constructed-response item rarely says "find f(4)"; it says "explain what f(4) = 90 means for the cost of the trip." Train yourself to translate symbols into sentences about the situation.
Functions also tie the whole exam together: lines, quadratics, exponentials, and sequences are all functions, and the same vocabulary — input, output, zero, intercept, increasing — reappears in every part. Mastering this section makes Chapters on quadratics and statistics far easier.
What Is a Function?
A function is a rule that assigns exactly one output to each input. The inputs form the domain; the outputs form the range.
The defining test: no input may produce two different outputs. An output can repeat (two inputs may share one output), but an input may never split.
- Mapping diagram: a function if each domain element has one arrow leaving it.
- Table: a function if no x-value is listed twice with different y-values.
- Graph: a function if it passes the vertical line test — no vertical line crosses the graph more than once.
A sideways parabola or a full circle fails the vertical line test, so neither is a function.
Function Notation and Evaluating
Function notation f(x) names a rule. The symbol f(x) is read "f of x" and stands for the output, not multiplication. To evaluate, substitute the input for every x.
Worked example. Let f(x) = 2x² − 3x + 1. Find f(−2).
- f(−2) = 2(−2)² − 3(−2) + 1
- = 2(4) + 6 + 1
- = 8 + 6 + 1 = 15
A common trap: forgetting that (−2)² = +4, or dropping the sign change in −3(−2) = +6. Substitute with parentheses every time.
Solving from notation. "Find x when f(x) = 0" means set the rule equal to 0 and solve — these x-values are the zeros (x-intercepts) of the function.
Domain and Range
Domain is the set of valid inputs; range is the set of outputs.
For most Algebra I lines and parabolas, the domain is all real numbers. The range narrows when the graph has a floor or ceiling: a parabola opening upward with vertex (1, −4) has range y ≥ −4.
Discrete vs. continuous. Context can restrict the domain. If x is the number of tickets sold, the domain is whole numbers (0, 1, 2, …), not all reals — you cannot sell 2.5 tickets. The Regents often asks for an "appropriate domain," so read the situation before answering.
Reading range from a graph. Scan the graph from the lowest point to the highest. A line that rises forever has range all reals; an upward parabola has range y ≥ (minimum); a downward parabola has range y ≤ (maximum). Always state range in terms of y-values, never x-values — mixing up the two is a frequent error.
Reading Key Features of a Graph
The Regents asks you to name and interpret these features in context:
| Feature | What to find | In context |
|---|---|---|
| y-intercept | value where x = 0 | starting amount / initial value |
| x-intercept (zero) | where graph meets x-axis, f(x)=0 | when a quantity equals zero |
| Increasing | interval where the curve rises left to right | quantity growing |
| Decreasing | interval where the curve falls | quantity shrinking |
| Maximum / minimum | highest or lowest output (vertex of a parabola) | greatest or least value reached |
| End behavior | direction of the graph as x → ∞ and x → −∞ | long-run trend |
Example. A ball's height h(t) peaks at (2, 80) then falls to a zero at t = 6. The maximum height is 80 ft at t = 2 s; the function increases on 0 < t < 2, decreases on 2 < t < 6, and the zero at t = 6 is when the ball lands.
End behavior describes what happens at the extremes. For a downward parabola, both ends point down (as x → ±∞, y → −∞). For an increasing exponential like y = 2^x, as x → ∞ the graph soars up, and as x → −∞ it flattens toward y = 0 (a horizontal asymptote). Describe end behavior with arrows or the phrase "approaches."
Which table does NOT represent a function?
If f(x) = x² − 5x + 2, what is f(−3)?