4.2 Relations, Functions, Function Notation & Function Families
Key Takeaways
- A relation is a function when every input maps to exactly one output; the vertical line test applies this rule graphically.
- Function notation f(3) = 7 means the input 3 produces the output 7; f(x) is never f times x.
- Domain is the set of permissible inputs and range the set of resulting outputs; division by zero and even roots of negatives are the two standard domain restrictions.
- Linear functions have constant first differences, quadratics have constant second differences, exponentials have constant ratios, and absolute value functions form a V shape.
- Square root functions start at an endpoint and increase without a mirror branch, while cube root functions extend through all real numbers in both directions.
4.2 Relations, Functions, Function Notation & Function Families
Skill 2 of Competency 2 is unusually explicit about which function types you must recognize: linear, quadratic, cubic, exponential growth and decay, absolute value, square root, and cube root. That list is a memorization target.
When is a relation a function?
A relation is any set of ordered pairs. It is a function when every input is paired with exactly one output.
- {(1, 4), (2, 7), (3, 4)} is a function. Two inputs share the output 4, which is fine — outputs may repeat.
- {(1, 4), (2, 7), (1, 9)} is not a function. The input 1 maps to both 4 and 9.
Graphically, this is the vertical line test: if any vertical line crosses the graph more than once, the relation fails. A circle fails; a parabola opening upward passes; a parabola opening sideways fails.
In a table, scan the input column for repeats, then check whether the repeated inputs carry the same output. In a mapping diagram, look for any input arrow that splits.
Function notation
f(x) names the output of function f at input x. f(x) does not mean f multiplied by x.
Given f(x) = 3x² − 5x + 2:
- f(4) = 3(16) − 5(4) + 2 = 48 − 20 + 2 = 30
- f(−2) = 3(4) − 5(−2) + 2 = 12 + 10 + 2 = 24
- f(a + 1) = 3(a + 1)² − 5(a + 1) + 2 = 3(a² + 2a + 1) − 5a − 5 + 2 = 3a² + a
Sign handling in f(−2) is where errors cluster: (−2)² = 4, not −4, and −5(−2) = +10.
Items also run notation backward. "For what x does f(x) = 0?" asks for the zeros, and "solve f(x) = 12" asks which inputs produce output 12.
Domain and range
Domain is the set of allowable inputs; range is the set of resulting outputs.
Two restrictions generate nearly every domain item:
- Denominators cannot be zero. For f(x) = 5/(x − 3), the domain is all real numbers except x = 3.
- Even roots require nonnegative radicands. For f(x) = √(x − 4), the domain is x ≥ 4. Odd roots have no restriction: the domain of ∛(x − 4) is all real numbers, because cube roots of negatives exist.
Context adds a third restriction. If n is the number of buses needed, the domain is nonnegative integers even though the algebraic rule accepts every real number. Items about discrete versus continuous domains are common in modeling contexts.
The function families
+---------------------------------------------------------------------------+
| FAMILY PARENT FORM SHAPE / SIGNATURE |
+---------------------------------------------------------------------------+
| Linear f(x) = x Straight line; constant 1st diffs |
| Quadratic f(x) = x^2 Parabola; constant 2nd diffs |
| Cubic f(x) = x^3 S-curve through origin; odd symmetry |
| Exponential f(x) = b^x Constant RATIO; horizontal asymptote |
| growth b > 1 rises left-to-right |
| decay 0 < b < 1 falls left-to-right |
| Absolute value f(x) = |x| V shape with a sharp vertex |
| Square root f(x) = sqrt(x) Half-parabola from an endpoint |
| Cube root f(x) = cbrt(x) S-curve, no endpoint, all reals |
+---------------------------------------------------------------------------+
Distinguishing features that items test directly:
- Quadratic vs. absolute value. Both are U- or V-shaped and symmetric. The parabola curves smoothly through its vertex; the absolute value graph has a sharp corner. Numerically, |x| has constant first differences on each side (it is piecewise linear), while x² does not.
- Quadratic vs. cubic. A parabola is symmetric about a vertical line and turns once; a cubic has rotational symmetry about a point and may turn twice or not at all. A cubic's third differences are constant.
- Exponential vs. quadratic growth. Both accelerate, but exponential growth has a constant ratio while quadratic growth has a constant second difference. Over the long run exponential always overtakes quadratic.
- Square root vs. cube root. √x stops at x = 0 and has a single branch in the first quadrant. ∛x continues into the third quadrant because negatives have real cube roots. This is the standard trap on domain items.
- Exponential decay vs. linear decrease. A linear decrease drops by a fixed amount each step and eventually goes negative; exponential decay drops by a fixed percent and approaches but never reaches zero.
Identifying a family from a table
x: 0, 1, 2, 3 and y: 3, 6, 12, 24. Ratios are 2, 2, 2 — exponential, y = 3 · 2ˣ. x: 0, 1, 2, 3 and y: 3, 6, 11, 18. Second differences are 2, 2 — quadratic. x: −2, −1, 0, 1, 2 and y: 5, 3, 1, 3, 5 — values decrease then increase symmetrically with constant slope magnitude 2 on each side — absolute value, y = 2|x| + 1.
That third case is worth practicing. Both quadratics and absolute value functions produce symmetric tables; only the absolute value function keeps the differences constant in magnitude on each side.
Given f(x) = 2x² − 7x + 3, what is f(−3)?
Which function has a domain of all real numbers?
A table shows x = -2, -1, 0, 1, 2 paired with y = 7, 4, 1, 4, 7. Which family best describes the relationship, and how can it be distinguished from the other symmetric family?