Change in tandem (function behavior)

AP Precalculus· Unit 1 · Polynomial & Rational Functions· 11 min read

1. What a Function Is: Domain, Range, and Variables★☆☆☆☆⏱ 3 min

Most interesting quantities change together. Fill a vase and the water's height rises as the volume poured in rises; the two vary in tandem. A function is the precise way we pin down that pairing: it takes each input and assigns it one — and only one — output.

📘 Definition

Function

f(x)

A relation that maps each input value to exactly one output value. If a single input could give two different outputs, the relation is not a function.

📘 Definition

Domain and range

The domain is the set of all input values the function accepts. The range is the set of all output values it produces.

The input is the independent variable (you choose it); the output is the dependent variable (its value depends on the input). We write $f (x)$ to mean the output the rule $f$ assigns to the input $x$ .

📘 Definition

Equal functions

Two functions $f$ and $g$ are equal when they have the same domain and, for every input $a$ in that domain, they give the same output: $f (a) = g (a)$ . Same formula but different domains means the functions are not equal.

📐 Worked Example

A stadium sells at most 4 tickets per person at $25 each. The total cost of $n$ tickets is $C (n) = 25 n$ . Identify the independent and dependent variables, and state the domain and range.

1
You decide how many tickets to buy, and the cost follows from that choice. So the input $n$ is the independent variable and the output $C$ is the dependent variable.
2
You can buy a whole number of tickets, from 0 up to the limit of 4. The domain is the set of allowed inputs:
3
$\text{domain} = \{0,\ 1,\ 2,\ 3,\ 4\}$
4
Apply the rule to each input to get the outputs $C (0) = 0, C (1) = 25, C (2) = 50, C (3) = 75, C (4) = 100$ . The range is the set of outputs:
5
$\text{range} = \{0,\ 25,\ 50,\ 75,\ 100\}$
6
Because only whole numbers of tickets make sense, both the domain and range are discrete sets, not intervals.

2. Four Representations, Image, and Preimage★★☆☆☆⏱ 3 min

The function rule — how each input is paired with its output — can be shown in four equivalent ways. They are four views of the same function, and a big part of this course is translating fluently between them.

📘 Definition

Four representations of a function

Graphical (a curve of points $(x, f (x))$ ), numerical (a table of input–output pairs), analytical (an equation such as $f (x) = x^{2}$ ), and verbal (a sentence describing the pairing).

📘 Definition

Image

The image of an input value $a$ is the single output the rule assigns to it, namely $f (a)$ . Every input in the domain has exactly one image.

📘 Definition

Preimage

The preimage of an output value $y$ is the set of all inputs the rule sends to $y$ — every $x$ with $f (x) = y$ . A preimage can contain no inputs, one input, or many.

📐 Worked Example

A function $f$ is given numerically by the table below. Find (a) the image of $- 2$ , (b) the preimage of $1$ , and (c) the preimage of $3$ .

$x$	$- 2$	$- 1$	$0$	$1$	$2$
$f (x)$	$4$	$1$	$0$	$1$	$4$

1
(a) The image of $- 2$ is simply the output paired with the input $- 2$ . Read the column under $x = - 2$ :
2
$f(-2) = 4$
3
(b) The preimage of $1$ is every input whose output is $1$ . Scan the bottom row for $1$ ; it appears under $x = - 1$ and $x = 1$ :
4
$\text{preimage of } 1 = \{-1,\ 1\}$
5
Notice the preimage is a set with two inputs — very different from an image, which is always one value.
6
(c) The preimage of $3$ is every input with output $3$ . No entry in the bottom row equals $3$ , so nothing maps to it:
7
$\text{preimage of } 3 = \{\ \}\ \text{(empty)}$

3. Increasing and Decreasing: Comparing Outputs★★☆☆☆⏱ 3 min

To describe how two quantities change in tandem, we first ask a purely qualitative question: as the input increases, does the output go up or down? We answer it by directly comparing output values — no measure of how fast is needed yet (that is the next topic).

📘 Definition

Increasing on an interval

A function is increasing on an interval of its domain if larger inputs always give larger outputs: for all $a$ and $b$ in the interval, if $a < b$ then $f (a) < f (b)$ .

📘 Definition

Decreasing on an interval

A function is decreasing on an interval of its domain if larger inputs always give smaller outputs: for all $a$ and $b$ in the interval, if $a < b$ then $f (a) > f (b)$ .

\text{increasing:}\ \ a<b \Rightarrow f(a)<f(b) \qquad \text{decreasing:}\ \ a<b \Rightarrow f(a)>f(b)

📐 Worked Example

A cup of tea cools. The table gives its temperature $T$ (°C) after $m$ minutes. On which interval is $T$ a decreasing function of $m$ ? Justify by comparing outputs.

$m$	$0$	$2$	$4$	$6$	$8$
$T$	$85$	$68$	$57$	$51$	$51$

1
Read the outputs from left to right and compare each with the next: $85 > 68 > 57 > 51$ .
2
From $m = 0$ to $m = 6$ , every larger input gives a strictly smaller output, which is exactly the definition of decreasing:
3
$0<2<4<6 \ \Rightarrow\ T(0) > T(2) > T(4) > T(6)$
4
So $T$ is decreasing on the interval $0 \leq m \leq 6$ .
5
From $m = 6$ to $m = 8$ the output stays at $51$ ( $T (6) = T (8)$ ), so on $6 \leq m \leq 8$ the function is constant, neither increasing nor decreasing.

4. Building the Graph: Concavity and Zeros★★☆☆☆⏱ 4 min

A graph is just the set of input–output pairs drawn in the plane, so it shows at a glance how the two values vary together. That means a verbal description of how two quantities change is already enough to sketch one — you do not need a formula first.

📐 Worked Example

Water is poured steadily into each of three vases: (a) one with straight sides, (b) one that gets wider toward the top, (c) one that gets narrower toward the top. For each, describe the graph of height $H$ against volume poured $V$ .

1
(a) Straight sides: every millilitre adds the same height, so the graph rises as a straight line.
2
(b) Wider toward the top: as the vase widens, each additional millilitre spreads out and adds less height than the one before. The graph still rises, but it flattens out as it goes.
3
(c) Narrower toward the top: each additional millilitre is squeezed into a smaller cross-section and adds more height than the one before. The graph rises ever more steeply.
4
All three are increasing — more water always means a higher level. What differs is how fast the height is changing, and that is what shapes the curve.

That last distinction has a name. Describing how fast the output changes is the subject of Topic 1.2; here we only need the idea of the rate of change getting larger or smaller as the input increases.

📘 Definition

Concave up

The graph of a function is concave up on intervals where the rate of change is increasing — each step of input raises the output by more than the step before. Vase (c) above is concave up.

📘 Definition

Concave down

The graph of a function is concave down on intervals where the rate of change is decreasing — each step of input raises the output by less than the step before. Vase (b) above is concave down.

One more feature is read straight off the picture: where the graph meets the horizontal axis.

📘 Definition

Zeros of a function

f(a)=0 \iff (a,\,0)\ \text{is on the graph}

The graph intersects the $x$ -axis exactly where the output value is zero. The corresponding input values are called the zeros of the function.

📐 Worked Example

Earlier we found that for $f (x) = x^{2} - 1$ the preimage of the output $0$ is ${- 1, 1}$ . What does that tell you about the graph?

1
The preimage of $0$ is the set of inputs whose output is $0$ — so $f (- 1) = 0$ and $f (1) = 0$ .
2
An output of $0$ puts the point on the $x$ -axis, so the graph passes through $(- 1, 0)$ and $(1, 0)$ .
3
Answer: the zeros of $f$ are $- 1$ and $1$ , and they are exactly where the graph crosses the $x$ -axis.

5. AP-Style Practice★★☆☆☆⏱ 5 min

✓ Quick check

Test your understanding with this multiple-choice question:

A function $h$ is given by the table below. On which interval is $h$ increasing?
$x$ $1$ $2$ $3$ $4$ $5$
$h (x)$ $12$ $7$ $4$ $7$ $12$
- $1 \leq x \leq 3$
- $3 \leq x \leq 5$
- $1 \leq x \leq 5$
- $2 \leq x \leq 4$
Reveal answer
1 —
From $x = 3$ to $x = 5$ the outputs rise: $4 < 7 < 12$ , so for any $a < b$ there, $h (a) < h (b)$ — increasing. On $1 \leq x \leq 3$ the outputs fall ( $12 > 7 > 4$ ), so that interval is decreasing, and the full interval $1 \leq x \leq 5$ mixes both.

$x$	$1$	$2$	$3$	$4$	$5$
$h (x)$	$12$	$7$	$4$	$7$	$12$

📐 Worked Example

Water is poured steadily into an empty vase. Let $H$ be the height of the water (cm) and $V$ the volume poured in (mL). (a) Which is the independent and which the dependent variable? (b) Explain why $H$ is an increasing function of $V$ . (c) The vase holds 500 mL and is 20 cm tall when full. State the domain and range of $H (V)$ .

1
(a) You control how much water goes in, and the height responds to that. So $V$ is the independent variable and $H$ is the dependent variable.
2
(b) Adding more water can only raise the level. So whenever $V_{1} < V_{2}$ , the taller column of water gives $H (V_{1}) < H (V_{2})$ . That matches the definition $a < b \Rightarrow f (a) < f (b)$ , so $H$ is increasing on its whole domain.
3
(c) The volume runs from empty to full, $0$ to $500$ mL, so the domain is $0 \leq V \leq 500$ .
4
The height correspondingly runs from $0$ to $20$ cm, so the range is $0 \leq H \leq 20$ .

📐 Worked Example

A function is described verbally: “the output is 1 less than the square of the input.” (a) Write it analytically. (b) Give the numerical representation for inputs $- 2, - 1, 0, 1, 2$ . (c) Find the preimage of the output $0$ .

1
(a) “Square of the input” is $x^{2}$ ; “1 less than” subtracts 1:
2
$f(x) = x^2 - 1$
3
(b) Evaluate the rule at each input to build the table:
4
$f(-2)=3,\ \ f(-1)=0,\ \ f(0)=-1,\ \ f(1)=0,\ \ f(2)=3$
5
(c) The preimage of $0$ is every input with output $0$ . From the table (or by solving $x^{2} - 1 = 0$ ), that is $x = - 1$ and $x = 1$ :
6
$\text{preimage of } 0 = \{-1,\ 1\}$

6. Common Pitfalls

Wrong move:

Reporting the preimage of an output as a single input value.

Why:

The preimage is the set of ALL inputs that map to the target output. For $f (x) = x^{2}$ , the preimage of $9$ is ${- 3, 3}$ , not just $3$ .

Correct move:

List every input whose output equals the target; the preimage may have zero, one, or several elements.

Wrong move:

Calling $x = y^{2}$ a function of $x$ .

Why:

The input $x = 4$ would give two outputs, $y = 2$ and $y = - 2$ , but a function must assign exactly one output to each input.

Correct move:

Check that each input yields only one output (graphically, that no vertical line meets the graph twice) before calling a relation a function.

Wrong move:

Reading the domain off the output side (or the range off the input side).

Why:

Domain and range get swapped: the domain is the set of inputs (independent variable), the range is the set of outputs (dependent variable).

Correct move:

Anchor on “inputs in, outputs out”: collect inputs for the domain and outputs for the range.

Wrong move:

Declaring $f (x) = \frac{x ^{2} - 4}{x - 2}$ and $g (x) = x + 2$ equal because the algebra simplifies.

Why:

$f$ is undefined at $x = 2$ while $g$ is defined there, so the two functions have different domains.

Correct move:

Functions are equal only when they share the same domain AND agree on every input; check the domains, not just the formula.

Wrong move:

Concluding $f$ is increasing on $[0, 4]$ just because $f (0) < f (4)$ .

Why:

Increasing requires $f (a) < f (b)$ for EVERY pair $a < b$ in the interval; two matching endpoints can hide a dip in between.

Correct move:

Verify the comparison holds across the whole interval, and never combine separate increasing/decreasing stretches into one claim.

7. Quick Reference Cheatsheet

Concept	Definition / Rule	Key Notes
Function	Each input maps to exactly one output	Fails if one input gives two outputs (vertical-line test)
Domain / Range	Domain = all inputs; range = all outputs	Domain feeds the rule; range comes out of it
Independent / Dependent variable	Input = independent; output = dependent	Output value depends on the chosen input
Equal functions $f = g$	Same domain AND $f (a) = g (a)$ for every $a$	Same formula but different domain ⇒ not equal
Four representations	Graphical, numerical, analytical, verbal	Four views of ONE rule; translate between them
Image of $a$	$f (a)$ — the single output for input $a$	Always exactly one value
Preimage of $y$	All inputs $x$ with $f (x) = y$	A SET: may be empty, one, or many
Increasing on $I$	For all $a < b$ in $I$ , $f (a) < f (b)$	Compare outputs; must hold for every pair
Decreasing on $I$	For all $a < b$ in $I$ , $f (a) > f (b)$	Compare outputs; must hold for every pair
Constant on $I$	$f (a) = f (b)$ for all $a, b$ in $I$	Neither increasing nor decreasing
Concave up	Rate of change is increasing	Each input step raises the output by more than the last
Concave down	Rate of change is decreasing	Each input step raises the output by less than the last
Zeros of a function	Inputs $a$ with $f (a) = 0$	Exactly where the graph meets the $x$ -axis

What's Next

You can now describe THAT a function's output rises or falls as its input increases, name a function's domain, range, image, and preimage across all four representations, and read concavity and zeros off a graph. The natural next question is HOW FAST it changes — that is Topic 1.2, Rates of Change, which measures the average rate of change over an interval and makes the idea of concavity precise.