What is a Function?
The definition the AP exam expects
Each input โ exactly one output
A function maps every input value to exactly one output value. This is the AP Precalculus definition.
Domain = set of inputs
The domain is represented by the independent variable (typically x or t). It's the set of all valid inputs.
Range = set of outputs
The range is represented by the dependent variable (typically y or f(x)). It's the set of all possible outputs.
Positive: y > 0 (above x-axis)
A function is positive where its graph lies above the x-axis โ outputs are greater than zero.
Negative: y < 0 (below x-axis)
A function is negative where its graph lies below the x-axis โ outputs are less than zero.
Don't say "Vertical Line Test"!
On the AP exam, you must use the statement "Each input has exactly one output" to explain why a relation is a function. The vertical line test is NOT accepted โ because in Unit 3, polar functions fail the vertical line test but ARE functions.
Four Representations
Every concept appears in all four forms
Graphical
Visual graph of the function โ most exam questions use this
Analytical
Equations like f(x) = 2x + 1 or f(x) = eหฃ + 1
Numerical
Tables of x and f(x) values
Verbal
Word descriptions of how inputs and outputs change together
The AP Exam tests all four
You need to identify information from any representation and construct equivalent representations โ for example, read a graph and write a verbal description, or read a table and determine increasing/decreasing behavior.
Graphical Behavior
The four behaviors every graph can show
Increasing
Outputs go UP as inputs increase
As x increases, f(x) always increases.
If a < b, then f(a) < f(b)
Decreasing
Outputs go DOWN as inputs increase
As x increases, f(x) always decreases.
If a < b, then f(a) > f(b)
Concave Up
Rate of change is INCREASING
The slope is getting steeper (more positive or less negative). Graph curves like a bowl โช.
Concave Down
Rate of change is DECREASING
The slope is getting less steep (less positive or more negative). Graph curves like a hill โฉ.
"Rate of change" = slope
Every time you see "rate of change," replace it with "slope" in your head. Concave up = slope increasing. Concave down = slope decreasing.
WORDS MATTER โ The Big Distinction
These two statements mean VERY different things:
โ
"f is increasing"
The outputs (y-values) are going up as inputs increase. The function itself is going up.
๐ "Rate of change of f is increasing"
The slope is getting bigger (more positive). This means the graph is concave up โ the function could even be decreasing! (e.g., decreasing but flattening out)
Reading Graphs โ Worked Examples
Examples 1โ4 from the notes
๐ Example 1 โ Water pours into a vase at a constant rate. The vase is wide at the bottom, narrow at the neck, then slightly wider at the top. Which graph shows height vs. time?
1
Wide base: diameter increasing โ water spreads out โ height rises at a decreasing rate โ concave down.2
Narrow neck: diameter decreasing โ water stacks up โ height rises at an increasing rate โ concave up.3
Height always increases (constant pour rate), never decreases. Start at 0 at t = 0.Answer: (D) โ always increasing, first concave down then concave up
๐ Example 2 โ Piecewise function f on [0, 9]. When is f increasing?
1
Look for sections where outputs go up as inputs increase โ the graph goes "uphill".2
From x = 0 to x = 3: graph rises. From x = 7 to x = 9: graph rises again.f is increasing on 0 < x < 3 and 7 < x < 9
๐ Example 3 โ Graph of h with F(โ6,4), G(โ3,0), J(0,โ4), K(3,0), P(6,4). tโ = t-coord of F = โ6, tโ = t-coord of G = โ3. On (tโ, tโ), which is true about h?
1
Interval is (โ6, โ3): h goes from 4 down to 0. Midline is y = 0 (the dashed line).2
Values: h starts at 4 > 0 and ends at 0 โ outputs are positive (above x-axis, above midline too).3
Going from 4 to 0: outputs decreasing.4
Rate of change: the graph is concave down on this interval โ rate of change is decreasing.(b) h is positive and decreasing | Rate of change is decreasing (concave down)
๐ Example 4 โ Same h, now K(9,โ4), P(12,โ1). tโ = 9, tโ = 12. On (tโ, tโ), which is true about h?
1
Interval (9, 12): h goes from โ4 up to โ1.2
Values: both โ4 and โ1 are below zero โ h is negative.3
Going from โ4 to โ1: outputs are going up โ h is increasing.4
This section curves like a hill (concave down) โ rate of change is decreasing.(c) h is negative and increasing | Rate of change is decreasing (concave down)
Graph of k โ Multiple Choice Examples
Examples 5โ8: practice identifying combined behaviors
๐ Graph of k on [0, 9]
๐ Segment Analysis โ Graph of k
| Section | Function Inc or Dec? |
Function Pos or Neg? |
Rate of Change Pos or Neg? |
Rate of Change Inc or Dec? |
|---|---|---|---|---|
| A โ B | โฌ๏ธ Increasing | Negative | Positive | โฌ๏ธ Increasing (concave up) |
| B โ C | โฌ๏ธ Increasing | Positive | Positive | โฌ๏ธ Decreasing (concave down) |
| C โ D | โฌ๏ธ Decreasing | Pos โ Neg (crosses x-axis) |
Negative | โฌ๏ธ Decreasing (concave down) |
| D โ E | โฌ๏ธ Decreasing | Negative | Negative | โฌ๏ธ Increasing (concave up) |
| E โ F | โฌ๏ธ Increasing | Negative | Positive | โฌ๏ธ Increasing (concave up) |
๐ Ex 5: On which interval is k negative AND decreasing?
โ
Look for a segment that is below the x-axis AND going downhill. That's DโE: k drops from โ1 to โ3.Answer: (C) D to E
๐ Ex 6: Which is true about the rate of change of k?
โ
EโF: the curve goes uphill (โ3 to โ1) โ slope is positive. Rate of change = slope.Answer: (D) Rate of change is positive on E to F
๐ Ex 7: Which is true about the rate of change of k?
โ
DโE: k is going downhill, but look at the shape โ it curves like a bowl (โช) โ concave up โ rate of change is increasing (slope becoming less negative).Answer: (C) Rate of change is increasing on D to E
๐ Ex 8: On which interval is k increasing AND concave down?
โ
BโC: the curve goes uphill (0 to 3) โ and arches over like a hill (โฉ) โ concave down โ.Answer: (B) B to C
Graphs f and g โ Interval Examples
Examples 9โ12
๐ Graph of f on [โ5, 5]
๐ Examples 9โ10: Graph of f on [โ5, 5].
9
On what interval is f decreasing? โ Find where the curve goes downhill.โ
f is decreasing on โ1 < x < 210
On what interval is f both negative AND increasing? โ Below the x-axis AND going uphill.โ
f is negative and increasing on โ5 < x < โ3๐ Graph of g on [โ5, 5] โ inflection point A(โ1, 0)
๐ Examples 11โ12: Graph of g on [โ5, 5]. A(โ1, 0) is the only inflection point.
11
On what interval is g decreasing AND concave up? โ Going downhill AND curving like a bowl โช.โ
g is decreasing and concave up on โ1 < x < 1 (after the inflection point, bowl shape)12
On what interval is the rate of change of g both positive AND decreasing? โ Slope is positive (going uphill) AND concave down (hill shape โฉ).โ
Rate of change is positive and decreasing on โ5 < x < โ3Quick Reference
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๐ Decision Guide
f is increasing โ outputs โ
Look at y-values going up or down
f is positive โ graph above x-axis
y > 0, NOT above the midline
concave up โ rate of change โ
Bowl shape โช โ slope is increasing
concave down โ rate of change โ
Hill shape โฉ โ slope is decreasing
Use input variable for intervals
Write intervals in terms of x (or t), never y
โ ๏ธ Common Mistakes
โ "Vertical line test"
Say "each input has exactly one output." Polar functions fail the VLT but are still functions.
โ "Positive" means above midline
Positive means above the x-axis (y > 0). Nothing to do with the midline.
โ Confusing f increasing with rate of change increasing
"f is increasing" = outputs go up. "Rate of change increasing" = slope increasing = concave up. A function can be decreasing AND concave up simultaneously.
โ Writing intervals in y-values
Always use the input variable (x or t) when writing intervals for features of a graph.