Chapter 1: Functions, Graphs, and Transformations#

Functions are the language calculus speaks. Before we can explore slopes, areas, or rates of change, we need a clear picture of what a function is and how to bend its graph to our will. This chapter builds that picture from the ground up.

The Big Picture#

Every pattern in nature — a ball’s height after you toss it, the temperature over a day, the population of a city — can be described by a function. A function is simply a rule that takes an input and produces exactly one output. Once we understand that rule, we can visualize it, shift it, stretch it, and combine it with others. These skills are the foundation of calculus, where we will soon ask “how fast?” and “how much?”. Right now, let’s focus on seeing and shaping the graphs themselves.

What Is a Function?#

Imagine a vending machine. You press the button for “A4,” and it drops a bag of chips. If you press A4 again, you get the same chips — not a candy bar. That’s a function: each input (button) gives a single, predictable output (snack).

Formally:

Function: A rule that assigns to each input exactly one output.

We usually write to mean “the output of the function when the input is .” The input is called the independent variable, and the output is the dependent variable because its value depends on .

A graph lets us see a function all at once. Every point on the graph is a pair . If a curve is the graph of a function, no vertical line can hit it more than once. Why? Because a single cannot produce two different ‑values. This is the vertical line test: a set of points is the graph of a function exactly when every vertical line crosses it at most once.

Domain and Range#

The domain of a function is the set of all inputs it can accept. The range is the set of all outputs it actually produces.

Domain: All possible ‑values for which the function makes sense. Range: All ‑values the function can output.

For , the domain is (you can’t take the square root of a negative number in the real numbers), and the range is . For , the domain is all real numbers except , and the range is also all real numbers except .

We can describe domain and range in words, with inequalities, or in interval notation. For example, the domain of is , written as .

Four Ways to Represent a Function#

We can pin down a function in four different ways, and each shines a different light on its behaviour.

1. Verbal – “Square the input, then add three.” This description is compact and human, but it can be ambiguous.
2. Tabular – A table of and values. It gives exact numbers but only for a few points.
3. Graphical – A picture of the curve. It reveals overall shape, peaks, valleys, and trends at a glance.
4. Algebraic – A formula like . It is precise and lets us calculate any output exactly.

In calculus, we often move between these representations: a verbal problem leads to an algebraic model, we graph it to see the big picture, and we use tables to check specific values.

📝 Section Recap: A function pairs each input with a single output, passes the vertical line test, and has a domain (allowed inputs) and a range (actual outputs). We can describe it verbally, with a table, a graph, or a formula.

Piecewise‑Defined and Absolute Value Functions#

Not every function follows a single neat formula for all inputs. A piecewise‑defined function uses different rules on different parts of its domain. Think of a tax bracket: you pay one rate on the first $50,000, a different rate on the next chunk.

A classic example is the absolute value function:

The graph of is a V‑shape with its point at the origin. For , the graph is the line ; for , it is the line (which slopes upward as we move left).

Piecewise functions let us model situations with sudden changes — a car that accelerates, then cruises, then brakes. To graph them, we draw each piece only over its specified interval and pay attention to whether endpoints are included (solid dot) or excluded (open dot).

Absolute value: The distance of a number from zero on the number line, always non‑negative.

The absolute value function can also be written as , which emphasizes that squaring removes the sign and the square root returns the positive distance. This perspective will be useful later when we differentiate it.

📝 Section Recap: Piecewise functions switch rules at different ‑values. The absolute value function is the simplest piecewise function; its graph is a V and it measures distance from zero.

Shifting Graphs: Moving Functions Around#

Once we know the graph of a basic function like , we can slide it around without changing its shape. These moves are called transformations, and they come from adding or subtracting numbers inside or outside the function.

Vertical Shifts#

Adding a constant to the output moves the graph up or down.

• shifts the graph up by units if , down if .

For example, is the standard parabola lifted 2 units upward. Every ‑coordinate gets added to it.

Horizontal Shifts#

Adding a constant to the input moves the graph left or right — but be careful: the direction feels “backwards.”

• shifts the graph right by units if , left if .

Why? Because to get the same output as before, must be larger by when we have . For , the vertex that was at now sits at — a rightward shift of 3.

A handy mental rule: changes inside the function (with ) affect the ‑coordinates in the opposite direction; changes outside affect directly.

📝 Section Recap: Vertical shifts add a constant to and move the graph up/down. Horizontal shifts add a constant to inside the function and move the graph left/right (opposite sign).

Stretching, Shrinking, and Reflecting Graphs#

Beyond sliding, we can squash, stretch, or flip a graph. These transformations change the scale or orientation.

Vertical Stretch and Shrink#

Multiplying the whole function by a constant scales the ‑values.

• : if , the graph stretches vertically (pulls away from the ‑axis). If , it shrinks vertically (squishes toward the ‑axis).

For , the wave peaks at and troughs at instead of and — three times taller.

Horizontal Stretch and Shrink#

Multiplying the input by a constant scales the ‑values, again with a “backwards” feel.

• : if , the graph shrinks horizontally (squeezes toward the ‑axis). If , it stretches horizontally (spreads away from the ‑axis).

For , the period is halved: the wave completes a full cycle twice as fast. For , the period doubles.

Reflections#

A negative sign flips the graph over an axis.

• reflects the graph across the ‑axis (top becomes bottom).
• reflects the graph across the ‑axis (left becomes right).

So is the square root curve flipped upside down, while is its mirror image on the left side of the ‑axis (domain ).

We can combine all these transformations. The general form packages several transformations: a vertical stretch/shrink by , a horizontal stretch/shrink by , a horizontal shift by , and a vertical shift by . When applying multiple transformations, a safe order is: horizontal shift, then horizontal stretch/shrink, then vertical stretch/shrink, then vertical shift. Or simply follow the “inside‑out” logic of the expression.

📝 Section Recap: Multiplying by a constant scales the ‑values (vertical stretch/shrink). Multiplying inside scales ‑values inversely (horizontal stretch/shrink). A negative sign reflects the graph over the ‑ or ‑axis.

Combining Functions: Building New from Old#

Functions are not isolated islands. We can add, subtract, multiply, divide, and even plug one into another to create richer models.

Arithmetic Combinations#

Given two functions and , we can form new functions by ordinary arithmetic, as long as we stay in the domain where both are defined.

• , provided

For example, if and , then with domain . The graph of the sum is simply the sum of the ‑coordinates at each .

Composite Functions#

A composite function feeds the output of one function directly into another. Think of an assembly line: machine A stamps out a part, and machine B paints it. The painted part is .

We write , which we read as “ of of .” The inner function is applied first, then works on the result.

Composite function: A function created by applying one function to the output of another: .

For instance, if and , then . Notice the domain is all real numbers because is always positive. If we reverse the order, , but only for . Order matters: is usually different from .

Decomposing Functions#

Often we need to break a complicated function into simpler pieces — the reverse of composition. This is called decomposition. For , we can see it as and , so . Decomposition is a creative skill: there is often more than one correct way, but the goal is to identify an “inside” function and an “outside” function.

For , a natural decomposition is (inside), (outside). In calculus, decomposition is essential for the chain rule, so we practice it early.

📝 Section Recap: We can add, subtract, multiply, or divide functions pointwise. Composition chains two functions together, applying the inner then the outer. Decomposition splits a complex function into inner and outer parts.

Summary#

We began with the simple idea that a function is a dependable rule: one input, one output. From there, we learned to see functions in four ways — words, tables, graphs, and formulas — and to talk about their domain and range. We met piecewise functions, including the V‑shaped absolute value, which change their rule mid‑stream. Then we discovered how to shift, stretch, and reflect graphs without changing their basic shape, giving us a toolkit to sketch almost any function quickly. Finally, we saw that functions can be combined with arithmetic or woven together through composition, and we practiced pulling them apart again. These skills are not just for graphing — they are the foundation on which all of calculus rests. Every derivative, every integral, every model starts with a function you understand deeply.