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Topic 1.1 · Function Definition
What is a function? (AP Precalculus answer)
NOT the vertical line test
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Topic 1.1 · Why Not VLT
Why is the vertical line test NOT accepted on the AP exam?
Think Unit 3
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Topic 1.1 · Domain
What is the domain of a function? What variable represents it?
Inputs or outputs?
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Topic 1.1 · Range
What is the range of a function? What variable represents it?
Independent or dependent?
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Topic 1.1 · Positive Function
When is a function f called 'positive'?
Above what line?
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Topic 1.1 · Negative Function
When is a function f called 'negative'?
Below what line?
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Topic 1.1 · 4 Representations
Name the four mathematical representations used in AP Precalculus.
Graph, ?, ?, ?
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Topic 1.1 · Increasing
Give the formal definition of an increasing function using inequality notation.
If a < b, then f(a) __ f(b)
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✓ Why Not VLT
Polar functions fail VLT but ARE functions
Unit 3 polar graphs fail the vertical line test yet are valid functions. The AP exam requires the statement each input has exactly one output.
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✓ Function Definition
Each input → exactly one output
A function maps every input value to exactly one output value. This is the AP-required language.
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✓ Range
Set of outputs · dependent variable
Range = all possible output (y) values. Represented by the dependent variable. Range of f⁻¹ = domain of f.
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✓ Domain
Set of inputs · independent variable
Domain = all valid input values. Represented by the independent variable (x or t). Always write intervals using the input variable.
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✓ Negative Function
f(x) < 0 — graph below the x-axis
Negative means below the x-axis (y < 0). Nothing to do with the midline!
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✓ Positive Function
f(x) > 0 — graph above the x-axis
Positive means above the x-axis (y > 0). ⚠️ NOT above the midline — a very common AP trap.
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✓ Increasing Definition
If a < b, then f(a) < f(b)
Outputs go UP as inputs increase. Formal: if a < b, then f(a) < f(b). The graph goes uphill left to right.
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✓ 4 Representations
Graphical · Analytical · Numerical · Verbal
Graphical = graph, Analytical = equation, Numerical = table, Verbal = words. AP exam tests all four — you must move between them.
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Topic 1.1 · Decreasing
Give the formal definition of a decreasing function using inequality notation.
If a < b, then f(a) __ f(b)
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Topic 1.1 · Concave Up
What does 'concave up' mean in terms of rate of change?
Graph shape: ∪
Rate of change = slope
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Topic 1.1 · Concave Down
What does 'concave down' mean in terms of rate of change?
Graph shape: ∩
Rate of change = slope
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Topic 1.1 · Key Distinction
What is the difference between 'f is increasing' and 'rate of change of f is increasing'?
One is about y-values, one is about slope
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Topic 1.1 · Vase Problem
Water fills a vase at constant rate. Wide base first, then narrow neck. Describe the height graph's concavity.
Wide → water spreads out. Narrow → water stacks up.
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Topic 1.1 · Intervals
When writing intervals where a function has a feature, which variable do you always use?
x-axis = input variable
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Topic 1.1 · Example 3
h: F(−6,4) → G(−3,0). t₁=−6, t₂=−3. On (t₁,t₂): is h positive or negative? Increasing or decreasing?
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Topic 1.1 · Example 8 (k)
Graph of k: B(2,0) → C(5,3). Is k increasing or decreasing? Is the graph concave up or down?
Going up from 0 to 3, but arching over
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✓ Concave Up
Rate of change is INCREASING
Concave up (bowl ∪) = slope is getting bigger (more positive or less negative). Rate of change = slope.
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✓ Decreasing Definition
If a < b, then f(a) > f(b)
Outputs go DOWN as inputs increase. Formal: if a < b, then f(a) > f(b). Graph goes downhill left to right.
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✓ Key Distinction
f increasing = outputs ↑ · Rate of change increasing = concave up
f is increasing: y-values go up.
Rate of change increasing: slope is increasing = concave up.
A function can decrease AND be concave up simultaneously!
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✓ Concave Down
Rate of change is DECREASING
Concave down (hill ∩) = slope is getting smaller (less positive or more negative). Rate of change decreasing = concave down.
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✓ Intervals use input variable
Always use x (or t) — the input variable
Write intervals like 2 < x < 5, not in terms of y. Features on a graph are described using input values.
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✓ Vase Problem
First concave down, then concave up
Wide base: water spreads → height rises at decreasing rate = concave down.
Narrow neck: water stacks → height rises at increasing rate = concave up.
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✓ Example 8: k on B→C
Increasing · Concave Down
B(2,0)→C(5,3): outputs go from 0 to 3 → increasing. Graph arches over the peak → concave down (rate of change is decreasing).
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✓ Example 3: h on (−6,−3)
Positive and Decreasing
F(−6,4)→G(−3,0): values go from 4 down to 0. Both ≥ 0 → positive. Going from 4 to 0 → decreasing.