Functions: Definition, Domain, Graphs, Transformations

This is Unit Five of College Algebra: functions. The capstone calls this the unit that installs the course's central object, and that is accurate. Everything in the four units after this one is a particular kind of function — polynomial, rational, exponential, logarithmic. This unit builds the general machine they are all instances of. The idea is a machine with one rule: each input produces exactly one output. Different inputs may share an output, and one input may never split. That single restriction is what makes everything else possible. Three named errors live here, and the first one is the most expensive mistake in the entire course — assuming a function distributes over a sum. Most functions refuse to, and the assumption survives because it looks like the distributive property, which is about multiplication. By the end you will be able to do five things. One: say why feeding in a sum is not the same as adding two outputs. Two: tell evaluating from solving, and say what each question asks. Three: find a domain by excluding what breaks the rule. Four: say why a subtraction inside the input moves a graph rightward. Five: find an inverse and confirm it by composition.

Four entries installing the object. A general notion first, then the restriction that turns it into the thing this course cares about, then the notation — which contains a trap in how it looks on the page.

The general notion.

Alg 231 Relations

Any collection of paired values. The first coordinates make up one set and the second coordinates the other — no restriction yet on how they pair up.

The restriction.

Alg 232 Functions

Each input leads to exactly one output. Note the asymmetry: several inputs may lead to the same output, and a single input may never lead to two.

The notation.

Alg 234 Function Notation

The name of the machine written beside its input. The whole thing is a single symbol naming an output — it is not one quantity multiplied by another.

Using it.

Alg 235 Evaluating a Function

Put the input in place of the variable everywhere it appears, then compute. Substitute inside brackets — the input goes in as a single block.

Three entries, and the middle one names the most expensive mistake in the course. It survives because it resembles a rule that is true — the distributive property — and that rule is about multiplication, which is not what is happening here.

Feeding in a sum.

Alg 236 Evaluating at a Sum

The machine receives the entire sum as one input. It is not opened up and fed in piece by piece.

And the error.

Alg 237 The Assumed-Linearity Error

Splitting the input and adding the two outputs. A small number of functions happen to permit it; almost none do, and assuming it is how correct working produces a wrong answer with no visible slip.

Two questions that look alike.

Alg 239 Evaluating versus Solving

Evaluating asks what a given input produces. Solving asks which inputs produce a given output. They run in opposite directions, and the second may have several answers or none.

A stronger condition, and two tests that read conditions straight off a picture. The two tests look identical in form and answer different questions — one asks whether you have a function, the other whether the function can be reversed.

The stronger condition.

Alg 241 One-to-One Functions

Each output traces back to exactly one input. The restriction that applied to inputs now applies in the other direction too.

The first test.

Alg 242 The Vertical Line Test

If any upright line meets the graph more than once, one input has two outputs, and it is not a function.

The second.

Alg 243 The Horizontal Line Test

If any flat line meets the graph more than once, one output belongs to two inputs, so it is a function and not a reversible one.

The check, on data.

Alg 245 Determining Whether a Relation Is a Function

Scan pairs, a table, or a graph for an input appearing twice with different outputs. Repeated outputs prove nothing at all — they are permitted, and looking at them is looking at the wrong column.

What is allowed in and what actually comes out. The first has a convention attached that does real work: unless something says otherwise, the domain is as large as the rule permits, so finding it means hunting for what breaks.

What may go in.

Alg 249 The Domain of a Function

Every permitted input. By convention it is as large as possible — every real number the rule can handle without failing.

What comes out.

Alg 250 The Range

Everything the allowed inputs actually generate. It is produced by the domain rather than chosen — you find it by working out what the machine can reach.

Finding it.

Alg 252 Finding a Domain from an Equation

Remove the inputs that break the rule — anything making a denominator zero, anything negative under an even root — and write what remains as intervals. Which bracket goes at each boundary is part of the answer.

Two entries. A function can use different rules on different stretches of its domain, and once that is available, one function you have used since Unit Three turns out to have been built that way all along.

Different rules, different regions.

Alg 254 Piecewise-Defined Functions

Several formulas, each owning a region of the domain, with the input deciding which applies. At every boundary, check which piece claims it.

The one you already knew.

Alg 256 The Absolute Value Function

Leave nonnegative inputs alone and flip the negative ones — two rules, two regions, and the corner is where they meet.

Three descriptions that need no formula at all. The first is Unit Three's slope generalised — the same calculation, applied where the graph is not a line, giving an average rather than a constant.

Slope, on a curve.

Alg 263 The Average Rate of Change

Change in output divided by change in input, between two points. On a line this is the slope; on a curve it is the average across that stretch.

Rising and falling.

Alg 265 Increasing and Decreasing Functions

Outputs climbing as inputs climb, or falling as inputs climb. These are described over stretches — a single point is not rising or falling.

The turning points.

Alg 266 Local Maxima and Minima

A point higher than everything immediately either side, or lower than both. These are where the rising and falling stretches change over.

Four operations that behave as you would expect, and then one that does not — because it is not an operation on outputs at all. It routes one machine's result into another machine's input slot, and order changes everything.

The four.

Alg 273 The Algebra of Functions

Add, subtract, multiply and divide the outputs at each input. For the quotient, the domain loses every input where the bottom function returns zero.

And the fifth.

Alg 275 Function Composition

One machine's result becomes another machine's input. The notation reads from the inside out — the inner one acts first.

Watch it evaluated.

Alg 276 Evaluating a Composite Function

Run the inner function on the input, then run the outer function on whatever came out. The notation is read right to left, which is the opposite of how it is written.

And why order matters.

Alg 278 Composition Order Matters

Composing one way, composing the other, and multiplying the two outputs are three separate operations giving three different results. None of them is a rearrangement of another.

Four entries on changing a graph without changing its shape. Changes made outside the function behave the way you expect. Changes made inside behave backwards — and that entry has a warning attached saying the sign lies, which is the honest description.

Outside, vertically.

Alg 281 Vertical Shifts

A constant added to the output lifts every point by that amount, and a negative one lowers it. This is the direction everybody expects.

Inside, horizontally.

Alg 282 Horizontal Shifts

A constant subtracted inside the input moves the graph to the right. It reads as though it should go left, and it does not — the sign is describing what the input must become, not where the graph goes.

Flipping.

Alg 284 Reflections

A negative applied to the output reflects across the horizontal axis; a negative applied to the input reflects across the vertical one.

And two symmetries.

Alg 285 Even and Odd Functions

A graph symmetric across the vertical axis returns the same output for a negated input; one symmetric through the origin returns the negated output. Run the test before claiming either.

Two more transformations, changing the shape rather than the position — and the inside-behaves-backwards rule from the last topic applies here too. Then a procedure, with an ordering rule that catches people.

Outside.

Alg 287 Vertical Stretches and Compressions

Multiplying the output by more than one stretches the graph vertically; a fraction compresses it. Any point sitting at zero output stays where it is.

Inside.

Alg 288 Horizontal Stretches and Compressions

Multiplying the input scales horizontally, and backwards again — a factor above one squeezes the graph, and one below stretches it.

Doing several at once.

Alg 290 Graphing by Transformations

Read the formula as edits to a parent graph — inside changes horizontal, outside changes vertical. On the outside, stretch before shifting: stretching an already-lifted graph lifts it further.

Five entries on reversing a machine. Only the reversible ones from Topic Three have inverses at all. There is a notation that misleads badly, and a test that settles whether two functions really are inverses without any guessing.

The reverse machine.

Alg 293 Inverse Functions

A function undoing another, sending each output back to the input it came from. Only one-to-one functions have one — otherwise the reverse would have to split.

The notation that misleads.

Alg 294 The Reciprocal-Inverse Error

The raised negative one names the reverse machine. It is not the reciprocal, and Unit One's negative exponent rule does not apply — that rule governs powers, and this symbol is not a power.

The test.

Alg 296 The Composition Test for Inverses

Compose them in both orders, and each composition must return the input untouched. Both directions have to work — one alone is not enough.

Finding one.

Alg 297 Finding an Inverse Algebraically

Write the output as a variable, exchange the two variables, solve for the new output, and confirm by composing. Swapping before solving is what performs the reversal — solving first inverts nothing.

And what it looks like.

Alg 299 Graphs of Inverse Functions

The inverse's graph is the original reflected across the diagonal line, because swapping each pair's coordinates is exactly that reflection.

The unit closes by naming what it installed and what the remaining units are — which reframes the rest of the course as four studies of one object rather than four separate subjects.

The capstone. The machine, installed.

Alg 300 The Function Machine

A machine giving every input exactly one output is the object the course is built on. Everything from here is machinery about particular kinds of it — and every claim in this entry carries an earlier number.

Five things you can now do. You can say why feeding in a sum is not adding two outputs — the sum enters as a single input, and the assumption that it splits resembles a rule about multiplication that does not apply here. You can tell evaluating from solving: one asks what a given input produces, the other asks which inputs produce a given output, and the second may have several answers or none. You can find a domain by removing what breaks the rule — zero denominators and negatives under even roots — and write the result as intervals with the right brackets. You can say why a subtraction inside the input moves a graph rightward: the sign describes what the input has to become, not where the graph travels. And you can find an inverse by swapping before solving, then confirm it by composing in both directions.