A vending machine, a taxi meter, a phone plan: in each one, a number you choose decides another number you get back. Mathematicians call these links relations, and the best-behaved ones functions. In this chapter you will learn to describe them with pairs, tables, mapping diagrams, graphs, and the famous notation \(f(x)\).
1. Relations, domain, and range
A relation is a set of ordered pairs \((x, y)\). The set of all first coordinates is the domain (the inputs). The set of all second coordinates is the range (the outputs).
A relation can be written as a list of pairs, shown in a table, drawn as a mapping diagram, or plotted as points on the coordinate plane. When you list the domain and range, write each value once and in increasing order.
Let \(R = \{(1, 4), (2, 6), (3, 6), (5, 9)\}\).
The first coordinates are \(1, 2, 3, 5\), so the domain is \(\{1, 2, 3, 5\}\).
The second coordinates are \(4, 6, 6, 9\). The value \(6\) appears twice but is listed once, so the range is \(\{4, 6, 9\}\).
2. What is a function?
A function is a relation in which every input has exactly one output. In other words, no \(x\)-value is ever paired with two different \(y\)-values.
Notice what the definition allows: two different inputs may share the same output. What it forbids is one input giving two different outputs, just like a vending machine that must give you exactly one snack for each button.
\(A = \{(2, 5), (3, 5), (4, 8)\}\) is a function: the inputs \(2, 3, 4\) each appear once. The repeated output \(5\) is fine.
\(B = \{(2, 5), (2, 7), (4, 8)\}\) is not a function: the input \(2\) is paired with both \(5\) and \(7\).
Students often reject a relation because an output repeats. That is allowed! Only a repeated input with different outputs breaks the rule.
3. Tables, mappings, and graphs
The same relation can wear several outfits. Here is the function that squares its input, for the inputs \(-2, -1, 1, 2\), as a table and as a mapping diagram.
| \(x\) | \(-2\) | \(-1\) | \(1\) | \(2\) |
|---|---|---|---|---|
| \(y\) | \(4\) | \(1\) | \(1\) | \(4\) |
In a mapping diagram, check the arrows leaving the input oval: if each input has one arrow, it is a function. Two arrows can land on the same output, but one input can never send out two arrows.
- List the inputs (the domain).
- Look for an input that appears more than once.
- If it does, compare its outputs: different outputs means not a function.
- If no input has two different outputs, it is a function.
4. The vertical line test
A graph represents a function if and only if no vertical line crosses it more than once.
Why does it work? A vertical line sits at one fixed \(x\)-value. If it touches the graph twice, that input has two outputs.
The circle fails: the line \(x = 2\) touches it at two points. The parabola below passes: every vertical line meets it at most once.
5. Function notation
Instead of writing \(y = 3x - 5\), we give the function a name, often \(f\), and write \(f(x) = 3x - 5\). Read it as “f of x”.
\(f(x)\) is the output of the function \(f\) when the input is \(x\). So \(f(4)\) means “the output when the input is 4”, and the point \((4, f(4))\) lies on the graph.
\(f(x)\) does not mean \(f\) times \(x\). The parentheses hold the input. Other letters work too: \(g(t)\), \(h(n)\), \(C(m)\).
On my planet we call a function a “machine”: drop an input in the top, and exactly one output falls out the bottom. Picture it whenever \(f(x)\) looks scary!
6. Evaluating functions
To evaluate a function, replace the variable by the given input (use parentheses!) and simplify.
Let \(g(x) = x^2 - 4x + 1\).
\(g(3) = 3^2 - 4(3) + 1 = 9 - 12 + 1 = -2\).
\(g(-1) = (-1)^2 - 4(-1) + 1 = 1 + 4 + 1 = 6\).
Let \(f(x) = 3x - 5\). Find \(x\) so that \(f(x) = 19\).
Solve \(3x - 5 = 19\), so \(3x = 24\) and \(x = 8\). Check: \(f(8) = 24 - 5 = 19\).
7. Graphs of functions
The graph of \(f\) is the set of all points \((x, f(x))\). To draw it, make a table of inputs and outputs, plot the points, and connect them when the domain is all real numbers.
| \(x\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|
| \(f(x) = 2x - 1\) | \(-3\) | \(-1\) | \(1\) | \(3\) |
Reading a graph goes the other way: to find \(f(2)\), start at \(x = 2\) on the horizontal axis, go up to the curve, then across to the vertical axis. The domain is the set of \(x\)-values the graph covers, and the range is the set of \(y\)-values it reaches.
8. Arithmetic sequences as functions
A sequence is a list of numbers; it is a function whose inputs are the positions \(n = 1, 2, 3, \dots\). In an arithmetic sequence, you add the same number \(d\), the common difference, to get from one term to the next.
If the first term is \(a_1\) and the common difference is \(d\), then \[ a_n = a_1 + (n - 1)d. \]
For \(5, 9, 13, 17, \dots\) we have \(a_1 = 5\) and \(d = 4\), so \(a_n = 5 + 4(n - 1) = 4n + 1\). The tenth term is \(a_{10} = 41\).
The graph is made of separate points, not a connected line, because \(n\) must be a positive whole number. The points lie on a straight line of slope \(d = 4\).
An arithmetic sequence has \(a_3 = 14\) and \(a_7 = 30\). Four steps separate them, so \(d = \dfrac{30 - 14}{4} = 4\). Then \(a_1 = 14 - 2 \cdot 4 = 6\) and \(a_n = 6 + 4(n - 1) = 4n + 2\). Check: \(a_7 = 28 + 2 = 30\).
Key takeaways
- A relation is a set of ordered pairs; the domain holds the inputs and the range holds the outputs.
- A function gives each input exactly one output. Repeated outputs are fine.
- Vertical line test: a graph is a function if no vertical line crosses it twice.
- \(f(x)\) is the output for input \(x\). To evaluate, substitute with parentheses.
- To solve \(f(x) = k\), set the rule equal to \(k\) and solve for \(x\).
- An arithmetic sequence is a function of \(n\): \(a_n = a_1 + (n - 1)d\).
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