Study guide: Functions, Nonlinear Patterns, and Domains
A function assigns one output to each allowed input. A table can reveal nonlinear structure or interval-average change, while the domain records which inputs make sense mathematically and in the supplied context.
Ideas to explain in your own words
- Repeated outputs are permitted in a function, but a repeated input cannot be paired with two different outputs. Inspecting input assignments tests the definition directly. The rule concerns one output per input, not one input per output.
- A quadratic table can arise by multiplying squared inputs by a constant. Its equal-input-step output differences change, which distinguishes its pattern from a linear relationship. Matching just one point cannot identify a rule.
- Average rate of change measures the net output change per net input change between two endpoints. It is the slope of the secant connecting those endpoint points. Keep subtraction order consistent in numerator and denominator.
- A counting context supplies input restrictions beyond a formula. Whole-item counts are integers, and a capacity constraint creates a largest allowed input while an empty collection may allow zero. Do not include fractions just because an algebraic formula can evaluate them.
Useful vocabulary
- function
- A relationship assigning exactly one output to each allowed input.
- domain
- The set of allowed input values.
- average rate of change
- Output change divided by input change over an interval.
Watch your reasoning
- The rule concerns one output per input, not one input per output. Group pairs by their first coordinate before judging the relation; two different inputs may legitimately share an output.
- Matching just one point cannot identify a rule. Check all supplied rows and compare first differences; the shown table supports the candidate formula among the stated options, not every conceivable formula outside the data.
- Keep subtraction order consistent in numerator and denominator. An interval average does not prove that the function changes at that rate at every intermediate input, particularly for a nonlinear function.
A worked review example
Complete supplied guide examples: Check Whether Every Input Has One Output — Among the supplied sets using inputs 2, 3 and 4, which defines y as a function of x? Candidate answers: (A) (2,1), (2,2), (3,3) (B) (2,1), (3,1), (4,2) (C) (3,0), (3,1), (4,2) (D) (4,2), (4,3), (2,1). The correct set assigns each of the three distinct inputs 2, 3 and 4 exactly one output. Repeated output 1 is allowed; one input with two outputs is not. Match Squared Inputs to a Table — At x=0,1,2,3 a rule has outputs 0,2,8,18. Which formula matches all four? Candidate answers: (A) y=2x (B) y=2x² (C) y=2x³ (D) y=x+2. The values are 2 times 0²,1²,2²,3². First differences are 2,6,10, not constant; this is not a linear rule. Calculate Average Change across an Interval — A function has f(2)=2 and f(4)=8. What is its average rate of change from x=2 to x=4? Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6. Use change in output divided by change in input: (8−2)/(4−2)=6/2=3. This is the interval average; it need not hold at every point. Restrict a Function to Meaningful Counts — A fictional box holds at most 7 whole markers. The input counts how many markers are in it. Which domain fits? Candidate answers: (A) All positive real numbers (B) All real numbers from 0 to 7 (C) Integers 0 through 7 (D) Integers greater than 7. You cannot have a fractional marker in this counting model, and an empty box is possible. The capacity makes 7 the largest allowed input.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.