Functions and their notation

A function is a rule that gives exactly one output for each permitted input. Two different inputs may give the same output. What cannot happen is a single input giving two different outputs.

The equation

Choose letters that fit the context. For example, $v(t)$ could represent speed $v$ in metres per second ($\text{m s}^{-1}$) at time $t$ in seconds ($\text{s}$). Similarly, $C(n)$ could represent a cost $C$ in the chosen currency for $n$ items, with $n$ as a unitless count. The notation shows what the input represents; it isn’t just decoration.

A domain is the set of inputs for which a function is defined. If no restriction is given, use the largest possible real domain. A range is the set of outputs actually produced by inputs from the domain. A graph of a function consists of the coordinate points whose first coordinate is an input and whose second coordinate is the corresponding output.

For example,

Functions as models

A mathematical model is a mathematical representation of selected relationships in a real situation. A function acts as a model once its input and output represent measurable or countable quantities. Its domain must satisfy the algebra as well as the context. For example, time may be limited to a recorded interval, while an item count may need to be a non-negative integer.

Temperature and currency conversions are naturally one-to-one functions, since every original value corresponds to one converted value. Cost functions link business decisions to expenditure. A projectile model can link time to quantities such as height. Each model simplifies reality, so its conclusions are only as reliable as the assumptions, data and chosen domain. Real situations motivate the model’s structure, and mathematics shows what follows within that structure.

One-to-one and inverse functions

A one-to-one function is a function where distinct inputs give distinct outputs. On its graph, every horizontal line intersects at most once.

An inverse function reverses the input-output assignment of a one-to-one function. It is written $f^{-1}(x)$, where $f^{-1}$ is the inverse of $f$ and $x$ is an input from the original range. The superscript $-1$ does not represent a reciprocal.

The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y=x$. As a result, the domain of $f^{-1}$ is the range of $f$, while the range of $f^{-1}$ is the domain of $f$.

If

This is why an inverse conversion takes a converted temperature or currency amount back to its original scale.

Representation and mathematical language

Function notation grew from the work of mathematicians in several European traditions before it became internationally standardized. Today, shared notation allows formulas to cross ordinary-language boundaries. In this limited way, mathematics works like a language, with symbols, syntax and conventions. But notation by itself isn’t mathematical knowledge. Meaning also comes from definitions, assumptions, proof and interpretation.

Reading the equation of a graph

The equation $y=f(x)$ gives the condition that every point on the graph satisfies: the vertical coordinate is the function value for the corresponding horizontal coordinate. The graph shows the same function as the algebraic rule, while making features such as turning points, restrictions and long-term trends easier to see.

A sketch is an approximate graph that keeps the important mathematical features but doesn’t require every point to be plotted exactly. A drawn graph is constructed to the accuracy required by the data, scale or method. A sketch can use a smooth freehand curve, but it must remain mathematically faithful—“not to scale” does not mean “features optional”.

When creating a sketch from algebra, written information or a practical context, label:

• both axes and any contextual units;
• the function or curve;
• any intercepts, endpoints and turning points supplied or found;
• asymptotes, discontinuities and relevant domain boundaries;
• the scale or significant coordinate values needed to interpret the graph.

The same method works for unfamiliar functions. Read the information given and identify the required features. Next, decide how the curve behaves between them. Only then should you join the pieces smoothly where continuity is appropriate.

Transferring a technology graph to paper

A calculator screen provides evidence; it isn’t the finished sketch. Start by choosing a window that shows the relevant behaviour. If the graph looks blank or misleadingly flat, check the domain, then adjust the horizontal and vertical ranges. Transfer the overall shape to paper and label its key features rather than trying to copy the screen pixels.

Technology is especially useful when graphing sums and differences. If $g$ is a second function with the same input variable as $f$, define

and

The domain of either new function contains only inputs that belong to the domains of both $f$ and $g$.

At each input, add the two original heights to get the height of the sum graph. Subtract them to find the height of the difference graph. The calculator carries out these vertical calculations across the window, but you still need to interpret the shape it produces.

Graphical and algebraic viewpoints

Graphs play a central role in sciences, geography and economics because their shape shows change, thresholds and trends quickly. Algebra is usually better for exact manipulation and justification. A graph, by contrast, makes global behaviour easier to see and allows solutions to be estimated. Neither representation is automatically more rigorous. Rigour depends on justifying claims and acknowledging limitations such as scale, rounding and viewing window. Using symbolic and visual forms together gives a fuller account than relying on either one alone.

Intercepts, zeros and roots

An intercept is a point where a graph meets one of the coordinate axes. An $x$-intercept has the form $(r,0)$

xxxFormulaEndxxx A $y$-intercept has the form $(0,q)$

xxxFormulaEndxxx Coordinates have no units unless the axes represent quantities in a context.

A zero of a function is an input where the function value equals zero. A root of an equation is a value that makes the equation true. For the equation $f(x)=0$, the roots are therefore the zeros of $f$. On a graph, they appear as the $x$-coordinates of the $x$-intercepts. Be precise with the terminology: a zero or root is usually an input value; an intercept is a point.

Maximum and minimum values

A maximum value is an output at least as large as every other output being considered. A minimum value is an output at most as large as every other output being considered. The chosen domain matters. Change the interval, and either value may change.

A local maximum is an output greater than or equal to the nearby outputs. A local minimum is less than or equal to nearby outputs. These local extrema don't have to be the highest or lowest points across the whole domain.

A vertex is the turning point of a parabola. It lies on the axis of symmetry and gives the parabola's maximum or minimum value.

Symmetry and asymptotes

A graph has line symmetry if reflecting it in a particular line leaves it unchanged. For symmetry about the vertical line $x=a$,

An asymptote is a line approached by a graph through its long-term or boundary behaviour. A vertical asymptote has equation

A horizontal asymptote has equation

The units of $L_v$ and $L_h$ match the units on their respective axes.

A graph can cross a horizontal asymptote, since a horizontal asymptote describes end behaviour. In contrast, a vertical asymptote marks an excluded input for the branch showing the unbounded behaviour.

Finding features with technology

Graphing technology can locate maxima, minima, zeros, intercepts and asymptotes, provided the feature appears in the displayed window. Start by graphing the function and selecting a suitable window. Then use the relevant zero, minimum or maximum command, recording the coordinates to the requested accuracy. To identify a vertical asymptote, inspect excluded inputs and unbounded behaviour. For a horizontal asymptote, examine increasingly large positive and negative inputs.

Sliders help when a formula includes a parameter. Change one parameter at a time, then watch which features move or alter. This visual test may suggest a relationship, but algebra or numerical evidence is still required before the pattern can be treated as established.

Intersections of two graphs

A point of intersection is a coordinate point lying on both graphs. For $y=f(x)$ and $y=g(x)$, intersections occur where

Plot both functions in the same window and use an intersection command near every visible meeting point. Report each result as a coordinate pair, not just an input value. Check the full relevant domain. Two curves can meet more than once, touch without crossing, or intersect outside the initial viewing window.

In an application, an intersection often shows that two modeled quantities are equal. For instance, the intersection of supply and demand curves represents market equilibrium. Features such as maxima and boundary points can help interpret production-possibility models. Similar graphical interpretation is used throughout experimental science and geography.

Analytical and visual approaches

The Bourbaki tradition is linked to highly formal, structural mathematics. Mandelbrot's work, by contrast, shows the insight that computational and visual exploration can produce. The two approaches complement each other rather than compete. A graph may show what needs investigation; analytical reasoning explains why the result occurs. The strongest conclusion usually comes from moving deliberately between both approaches.

Composite functions

A composite function uses the output of one function as the input of another. The notation

Read the expression from right to left. In general, $f\circ g$ and $g\circ f$ are different functions.

The domain of $f\circ g$ consists precisely of the inputs permitted by $g$ whose outputs can then be used as inputs to $f$. In context, this check matters. One function might convert a measured quantity into another unit, for example, while a second calculates a charge from the converted value. The order matches the actual sequence of operations.

Inverses and domain restriction

An inverse must also be a function, so the original function needs to be one-to-one on its stated domain. If a function fails the horizontal-line test, it may become one-to-one after a domain restriction—a deliberate replacement of its domain with a smaller set of permitted inputs.

For example,

is not one-to-one over all real numbers. Restricting its domain to $x\ge -1$ selects the increasing branch. On this domain,

Its domain starts at $4$, the range boundary of the selected branch. A restriction to $x\le -1$ would instead select the other branch and give a different inverse.

Finding an inverse algebraically

To find an inverse function:

1. Write the function as $y=f(x)$.
2. Interchange $x$ and $y$, since the inputs and outputs exchange roles.
3. Rearrange to make $y$ the subject.
4. Rename the result $f^{-1}(x)$.
5. State any required domain restriction and swap the original domain and range.

Sometimes rearranging produces more than one branch. Don’t automatically keep both. Use the restricted domain of the original function to select the branch that makes the inverse a function.

Composition with an inverse

Applying a function and then its inverse restores the original value. On the appropriate domains,

and

The first identity applies when $x$ belongs to the range of $f$, which is the domain of $f^{-1}$. The second applies when $x$ lies in the domain on which $f$ is one-to-one. To check a proposed inverse directly, compose in both orders and verify the algebra and the domains.