Relations, Functions and Graphs I

A relation links one set of values to another. A function is a relation with one extra rule, and that rule is what makes the whole of the rest of mathematics work.

Relations and functions

A relation can be given as a set of ordered pairs, a mapping diagram, a table, a graph or a rule.

$$\{(1,3),\ (2,5),\ (4,9)\}$$

A relation is a function when every input has exactly one output.

$\{(1,2),(2,4),(3,6)\}$ is a function — each of 1, 2 and 3 leads to a single value.

$\{(1,3),(2,5),(1,7),(4,9)\}$ is not a function, because the input 1 leads to both 3 and 7.

The vertical line test

On a graph, a relation is a function if no vertical line crosses the curve more than once. A vertical line represents a single $x$ value, so two crossings would mean one input with two outputs.

A circle fails the test. A straight line (that is not vertical) passes it. A parabola $y = x^2$ passes it.

Domain and range

The domain is the set of inputs. The range is the set of outputs actually produced.

For $\{(1,3),(2,5),(4,9)\}$:

• Domain: $\{1, 2, 4\}$
• Range: $\{3, 5, 9\}$

Find the range of $f(x) = x^2$ for the domain $\{-2, -1, 0, 1, 2\}$.

Square each input: $4, 1, 0, 1, 4$. A set does not repeat values, so the range is $\{0, 1, 4\}$.

Function notation

$f(x)$ is read "f of x" and names the output of the function $f$ for the input $x$. It does not mean $f$ multiplied by $x$.

To evaluate, replace every $x$ with the given value.

If $f(x) = 2x + 7$, find $f(3)$.

$$f(3) = 2(3) + 7 = 6 + 7 = 13$$

If $f(x) = 5 - 2x$, find $f(-3)$.

$$f(-3) = 5 - 2(-3) = 5 + 6 = 11$$

You can also work backwards. If $f(x) = 4x - 3$ and $f(x) = 9$, find $x$.

$$4x - 3 = 9 \quad \Rightarrow \quad 4x = 12 \quad \Rightarrow \quad x = 3$$

Composite functions

$fg(x)$ means: apply $g$ first, then apply $f$ to the result. Work from the inside out, exactly as with brackets.

Let $f(x) = 3x - 5$ and $g(x) = x + 2$.

$$fg(x) = f(x+2) = 3(x+2) - 5 = 3x + 6 - 5 = 3x + 1$$

Going the other way:

$$gf(x) = g(3x-5) = (3x - 5) + 2 = 3x - 3$$

Check numerically at $x = 3$: $fg(3) = 3(3) + 1 = 10$, and going step by step, $g(3) = 5$ then $f(5) = 10$ ✓. Meanwhile $gf(3) = 3(3) - 3 = 6$ — a different answer, as expected.

Inverse functions

$f^{-1}(x)$ undoes $f$. If $f$ turns 4 into 7, then $f^{-1}$ turns 7 back into 4.

To find it: write $y = f(x)$, make $x$ the subject, then swap the letters.

Find the inverse of $f(x) = 3x - 5$.

$$y = 3x - 5$$ $$y + 5 = 3x$$ $$x = \frac{y+5}{3}$$ $$f^{-1}(x) = \frac{x+5}{3}$$

Check: $f(4) = 3(4) - 5 = 7$, and $f^{-1}(7) = \dfrac{7+5}{3} = 4$ ✓ — back where we started.

Solving equations graphically

Where two graphs cross, both equations are satisfied at once — so the point of intersection is the solution of the simultaneous equations.

The lines $y = 2x + 1$ and $y = 7 - x$ intersect. Find the point.

Algebraically, set them equal:

$$2x + 1 = 7 - x$$ $$3x = 6$$ $$x = 2$$

Then $y = 2(2) + 1 = 5$.

The lines cross at $(2, 5)$.

Reading a solution off a graph you have drawn gives an estimate, and the accuracy depends on your scale. If a question says "use your graph", an answer to one decimal place is normally expected, and you should say which graph you read it from.