Functions

Function Notation #

f(4) means put 4 in, not f times 4.

A function is a machine that turns each input into exactly one output

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f(x) names a rule. Whatever goes in, the rule says what comes out.

f(4) does not mean f times 4. It means substitute 4 for x: 3 × 4 + 1 = 13.

Plot every input against its output and the rule becomes a picture.

Working backward, g(x) = 10 asks which inputs give 10: both x = 3 and x = −3 do.

Domain and Range #

What may go in, and what can come out.

The domain is what you are allowed to put in and the range is what can come out

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Any number can be squared, so every number is an allowed input. The set of allowed inputs is the domain.

But $x^2$x² is never negative, so the range, the set of outputs, is every number from 0 up.

Here x cannot be zero at all, because dividing by zero has no answer.

And a square root needs what is inside to be 0 or more, so its domain starts at 0.

Piecewise Functions #

One rule for some inputs, another for the rest.

A piecewise function gives a different rule on each interval of its domain

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One function can carry two rules, each with the inputs it governs.

To evaluate, first find which interval the input is in, then use that rule.

x = 8 is in the other interval, so use the second rule: 10 − 8 = 2.

The boundary belongs to whichever rule carries the "or equal to".

Graphing a Piecewise Function #

Filled where the point is in, hollow where not.

Each rule is drawn only over its own interval, and the endpoints show which piece includes the boundary

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Two rules again, and this time they disagree about x = 2.

Draw each rule only over its own interval, and the graph breaks at x = 2.

A hollow dot marks a point that is not there; a filled dot marks one that is.

The strict < leaves that end open; the $\ge$≥ closes the other one.

Change the second rule to 5 − x and both give 3 at x = 2, so the break closes.

Making a Piecewise Function Continuous #

Choose the constant that closes the jump.

The two pieces join without a jump exactly when both rules agree at the boundary

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f(x) = 2x for x < 2 and x + 4 for $x\ge 2$x ≥ 2: the graph jumps by 2 at the boundary.

Replace the 4 with an unknown k, and ask which value of k closes the jump.

Work out both rules at the boundary: one gives 4, the other gives 2 + k.

Set the two equal, and one value of k is left: k = 2.

With k = 2 both rules give 4 at x = 2, so the graph has no break: f is continuous.

Composite Functions #

One machine fed into the next, right to left.

Feeding one function into another makes a composite, and the order changes the answer

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Here are two functions: f adds 3, and g doubles.

gf means do f first: 4 becomes 7, then doubling gives 14.

fg means do g first: 4 becomes 8, then adding 3 gives 11. The order matters.

Inverse Functions #

Reverse the steps, and reverse each one.

The inverse undoes a function by running every step backwards

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Read the rule as steps in order: multiply by 3, then add 1.

To undo it, reverse the order of the steps and swap each step for its opposite.

The inverse is the original reflected in the line y = x.

Restricting the Domain for an Inverse #

Cut a curve at its turning point so it can be undone.

A function has an inverse only where each output comes from one input, so restrict its domain until it does

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$y=x^2$y = x² gives 4 at both x = −2 and x = 2, so an inverse would not know which one to return.

Keep only $x\ge 0$x ≥ 0 and every horizontal line meets the curve at most once: f is now one-to-one.

On $x\ge 0$x ≥ 0 the inverse is $f^{-1}\left(x\right)=\sqrt{x}$f⁻¹(x) = √x, the reflection of that half in the line y = x.

For ${\left(x-2\right)}^2+1$(x − 2)² + 1, start the domain at the vertex, $x\ge 2$x ≥ 2, and take the positive root.

When a Composite Function Exists #

The range of g has to fit inside the domain of f.

fg exists only when every output of g is an input f accepts: the range of g lies inside the domain of f

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fg feeds the outputs of g into f. With $f\left(x\right)=\sqrt{x}$f(x) = √x and g(x) = x − 3, g gives −2 at x = 1, and $\sqrt{-2}$√(−2) does not exist.

So fg exists only when the range of g lies inside the domain of f. On $x\ge 3$x ≥ 3, g gives $y\ge 0$y ≥ 0, which fits.

Then fg$\left(x\right)=\sqrt{x-3}$(x) = √(x − 3), and its domain is the domain of g, $x\ge 3$x ≥ 3, because x goes into g first.

The order matters. gf feeds the outputs of f, $y\ge 0$y ≥ 0, into g, which takes every number, so gf exists.

Transforming Graphs #

Outside the bracket moves it the way you expect.

Adding outside the bracket moves the graph up or down; adding inside moves it sideways, against the sign

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Start with $y=x^2$y = x², whose vertex is at the origin.

Add 3 outside the function and every point moves up 3.

Subtract 3 inside the bracket, ${\left(x-3\right)}^2$(x − 3)², and the curve moves 3 to the right.

x must be 3 larger to give the same value inside the bracket, so every point moves 3 to the right.

A line follows the same rule. Moved 2 right, y = 2x + 1 becomes y = 2x − 3: the intercept falls by 2 × 2.

Stretching a Graph Vertically #

Outside the bracket, every height multiplies.

Multiplying a function from outside the bracket multiplies every height by the same factor

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$y=2x^2$y = 2x² doubles every height, so the curve is stretched away from the x-axis.

At x = 2 the height goes from 4 to 8. The heights double, and no point moves sideways.

A factor less than 1 squashes it toward the x-axis, by the same rule.

A negative factor sends every height to the other side: $y=-x^2$y = −x² is the reflection of $y=x^2$y = x² in the x-axis.

Points on the x-axis have height 0, so they do not move, while every other point is stretched.

Stretching a Graph Horizontally #

Inside the bracket, the widths divide instead.

Replacing x by ax inside the bracket squashes the graph toward the y-axis by that factor

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Start from $y=x^2-4$y = x² − 4, which meets the x-axis at −2 and at 2.

Replace x by 2x, and the inside of the bracket reaches each old value at half the x.

So $y={\left(2x\right)}^2-4$y = (2x)² − 4 crosses at −1 and 1: the curve is squashed toward the axis.

Replacing x by $\frac{x}{2}$x/2 stretches it outward instead, to −4 and 4.

Outside the bracket multiplies the height; inside it divides the width.

Combining Graph Transformations #

Two changes in a row, and the order decides where a point lands.

Follow one point: outside the bracket the height goes through the arithmetic in order; inside, x goes through it backwards

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y = 2f(x) + 1 changes y = f(x) twice. Here $f\left(x\right)=\sqrt{x}$f(x) = √x, and the point (4, 2) ends at (4, 5).

Outside the bracket, follow the arithmetic in order: the height is doubled first, and 1 is added after.

The order matters: translating first and stretching after gives 2f(x) + 2, not 2f(x) + 1.

Inside the bracket, work backwards: x = 0 makes 2x + 4 = 4, so the point at x = 4 moves to x = 0.

So f(2x + 4) is a translation 4 to the left, followed by a stretch of scale factor ½ parallel to the x-axis.

Exponentials and Logarithms #

A log asks what power the base was raised to.

A logarithm asks what power the base was raised to, so it undoes an exponential

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$y=2^x$y = 2ˣ doubles each time x goes up by 1, so it rises slowly at first, then very fast.

The same fact, read the other way: what power of 2 gives 32?

The graph of $y={\log }_{2}x$y = log₂ x is the reflection of $y=2^x$y = 2ˣ across the line y = x: the domain and range change places.

Exponential Growth #

Multiplying by the same factor every step.

Exponential growth multiplies by the same factor every step, so it accelerates

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Doubling each step: 1, 2, 4, 8, 16. Each jump is as big as everything before it.

Adding the same amount each time gives a line. Multiplying gives this curve.

So after n steps the count is the start times $2^n$2ⁿ — with ×3 each step, times $3^n$3ⁿ.

Exponential Decay #

The same rule with a factor below one.

The same rule with a factor below one shrinks toward zero without reaching it

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Halving each step: 80, 40, 20, 10, 5. The same rule, with a factor below 1.

The curve falls steeply, then flattens as there is less left to halve.

It closes in on zero forever without reaching it — half of something is never zero.

Growth and Decay Problems #

Doubling colonies and half-lives, one rule twice.

Repeated doubling and half-lives are the two exponential shapes at work

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Doubling each hour is exponential growth: after n hours, 100 has become $100\times 2^n$100 × 2ⁿ.

A half-life of 3 days: 9 days is three halvings, so 80 g drops to 40, 20, then 10.

A savings account grows this way, a cooling coffee cools this way — the same two shapes.

Logistic Growth and Carrying Capacity #

Growth that slows as the room runs out.

A logistic model grows like an exponential at first and then flattens onto the largest size the space allows

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Exponential growth never lets up. Doubling forever would need unlimited room.

A real population flattens onto a ceiling instead: the carrying capacity.

As t grows the exponential fades to nothing, so P closes on L and stops.

At the start the same formula hands back the starting size, L over 1 + A.

Growth is fastest halfway up, then slows as the remaining room runs out.

The Laws of Logarithms #

A log turns multiplying into adding.

A logarithm turns multiplying into adding, which is what makes it useful

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The law of exponents: multiplying powers of one base adds the exponents, $2^3\times 2^4=2^7$2³ × 2⁴ = 2⁷.

So log(ab) = log a + log b, in any base — so long as it is one base throughout.

And $\log \left(x^n\right)=n\log x$log(xⁿ) = n log x: a power is repeated multiplying, so the n comes to the front.

And $\log \left(a/b\right)=\log a-\log b$log(a / b) = log a − log b: dividing inside becomes subtracting outside.

The Change of Base Rule #

Any log is a quotient of logs you can compute.

One rule turns any logarithm into a quotient of logs the calculator has

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This x is ${\log }_{2}N$log₂ N: applying the common logarithm isolates it, through the change-of-base identity.

So ${\log }_{2}N$log₂ N is $\log N/\log 2$log N / log 2 — and any base converts the same way.

A calculator offers only ${\log }_{10}$log₁₀ and ln — this rule reaches every base from those two.

So ${\log }_{5}30$log₅ 30 is 2.113 — and 5 to that power really does come back to 30.

The Natural Logarithm #

The log with base e, partner of e to the x.

ln is the logarithm with base e, and it obeys every law the others do

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ln is the log with base $e\approx 2.718$e ≈ 2.718, the base that turns up wherever growth is continuous.

$\ln e^x=x$ln eˣ = x: whatever power e was raised to, ln hands it straight back.

The other way round too: $e^{\ln x}=x$e^(ln x) = x. Each machine undoes the other.

Every law of logarithms carries over unchanged — ln is a log like any other.

Solving Exponential Equations #

Take logs and the power comes down.

Taking logs of both sides brings the unknown power down where you can reach it

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The unknown is an exponent, so no amount of dividing will free it.

Take the log of both sides, and the power law brings the x down from the exponent.

Now it is an ordinary division. Check it: 2 to the 5 really is 32.

And here counting could never have got there: 40 is not any whole power of 2.

Logarithmic Graphs #

The exponential reflected in y = x.

The log graph is the exponential reflected in the line y equals x

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$y=2^x$y = 2ˣ climbs away steeply and never dips below the x-axis.

Undoing swaps input and output, and swapping x with y reflects a graph in y = x.

So $y={\log }_{2}x$y = log₂x is $y=2^x$y = 2ˣ reflected in that line: every (a, b) becomes (b, a).

So it dives at 0 and never reaches it: no power of 2 will ever give zero.

Straightening Growth with Logarithms #

Take logs and the curve becomes a readable line.

Taking logs turns an exponential law into a straight line whose rule can be read

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$y=3\times 2^x$y = 3 × 2ˣ sweeps up in a curve — its 3 and its 2 are hard to read off the picture.

Take logs of both sides: log y = log a + x · log b — a straight line in x.

Plot log y instead of y and the curve straightens: same data, now a line.

Undo the logs: the gradient gives b = 2 and the intercept a = 3 — the rule is recovered.

Straightening a Power Law #

On log-log axes the slope is n itself.

A power law straightens on log-log axes, and the slope is n itself

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$y=2x^3$y = 2x³ sweeps up in a curve — is the power 2, 3 or 4? The picture will not say.

Take logs: log y = log a + n · log x — a straight line in log x, not in x.

Plot log y against log x and it straightens — same data, and the slope is n itself.

Read the rule off: n = 3 from the slope, and 10 to the intercept gives a = 2.

Fitting a Logarithmic Model #

Rising forever, by less and less each step.

A logarithmic model y = a + b ln x keeps rising while each extra unit of x adds less than the last

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Scores climb with practice, but each extra trial adds less than the one before.

A logarithmic model y = a + b ln x bends that way: it rises without a ceiling.

Plot the same scores against ln x and the bend straightens into a line.

Read a off as the intercept of that line and b off as its gradient.

Put a value in and the model predicts: twenty trials gives a score near 17.

Which axis you take logs of is what picks the family of model.