Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7: a function machine doubles the input and then adds 3. What does it output for the input 5?
Answer
13. Double 5 to get 10, then add 3. - From Year 9: a line has gradient 2 and passes through (0, −1). What is its equation?
Answer
y = 2x − 1. The gradient is the number in front of x, and the y-intercept is −1. - Work out 3 × (−4)² − 5.
Answer
43. Square first: (−4)² = 16, then 3 × 16 = 48, then 48 − 5 = 43.
Words used in this lesson
- function
- a rule that gives every input exactly one output. For example: the cost of a parcel from its weight.
- domain
- the set of inputs a function can accept. For example: temperatures from 0 to 40.
- range
- the set of outputs a function actually gives. For example: fares from 8 up to 90.
- relation
- any set of ordered pairs, whether or not it is a function. For example: {(1, 2), (1, 3)}.
- vertical line test
- a graph is a function if no vertical line meets it twice. For example: a circle fails it.
- f(x)
- the output of the function f for the input x. For example: f(2) means put 2 in place of x.
One card, one borrower
A library gives every card a number and writes a borrower's name against it. Some borrowers hold more than one card. Rules like this appear everywhere: a bus fare from a distance, a price from a weight, a score from a time.
Predict first: is “card number → borrower” a function? Is “borrower → card number” a function? Then use the figure to see how a graph shows the difference.
y = 2x − 1 for 0 ≤ x ≤ 3: the line x = 1 meets it once, at y = 1. No vertical line meets it twice, so it is a function.
Show me the steps
- Press y = 2x − 1 and slide the line to x = 2. It meets the graph once, at y = 3. The domain is 0 ≤ x ≤ 3 and the range is −1 ≤ y ≤ 5.
- Press x = y² and slide to x = 4. The line meets it twice, at y = −2 and y = 2. One input has two outputs, so it is not a function.
- Press x² + y² = 16, a circle. The line x = 0 meets it at y = −4 and y = 4; the line x = 4 meets it once, at y = 0. Domain and range are both from −4 to 4.
- Press y = √x and slide to x = −2. The line misses it, because −2 is outside the domain (0 ≤ x ≤ 4). The range is 0 ≤ y ≤ 2.
A function gives every input exactly one output. The set of inputs allowed is the domain; the set of outputs that actually occur is the range. On a graph, a relation is a function when no vertical line meets it more than once.
Function notation. f(x) = 3x − 2 names the rule f and says “the output for input x”. f(4) means put 4 in place of every x. The graph of the function is the set of points (x, f(x)).
Two inputs may share an output, as 3 and −3 do under y = x². An input may never have two outputs.
Two common mistakes. f(x) is not f multiplied by x: it is one symbol for the output. And a negative input needs brackets: if f(x) = x² + 2, then f(−3) = (−3)² + 2 = 11, not −3² + 2 = −7.
Worked examples
1. Is {(0, 2), (1, 3), (4, 3), (6, 7)} a function? State its domain and range.
- List the inputs (the first numbers): 0, 1, 4 and 6. Each appears once, so each has one output.
- It is a function. The domain is {0, 1, 4, 6}.
- The outputs are 2, 3, 3 and 7. List each value once: the range is {2, 3, 7}. Two inputs share the output 3, which is allowed.
2. f(x) = 2x² − 3x + 1. Find f(−2).
- Replace each x by (−2), with brackets: 2(−2)² − 3(−2) + 1.
- (−2)² = 4, so the first term is 8, and −3 × (−2) = +6.
- f(−2) = 8 + 6 + 1 = 15.
3. State the domain and range of g(x) = √(x − 1).
- A square root needs a number that is not negative: x − 1 ≥ 0, so the domain is x ≥ 1.
- A square root is never negative, and it reaches every value from 0 upward: the range is g(x) ≥ 0.
Your go. The first step is given. g(x) = 10 − x². Find g(−3).
- Put (−3) in place of x, with brackets: g(−3) = 10 − (−3)².
- Now square first, then subtract.
Check the answer
(−3)² = 9, so g(−3) = 10 − 9 = 1. Without the brackets, −3² would be −9 and the answer would be 19.
With a GDC (optional)
A GDC can tabulate a function for you. Enter the rule in the graph menu's list of functions and open the table of values: the column of outputs is f(x) for every whole-number input. For f(x) = x³ + 2x the table shows 12 beside x = 2, which matches 8 + 4 done by hand.
Use the table to check, not to think. A graph window only shows part of a function, so a graph that seems to stop is not evidence about the domain. Set the window yourself and say in words what you see. Everything here can be done by hand.
Check before you go on
1. Which set of ordered pairs is a function?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. f(x) = x² − 4x + 7. Find f(−3).
3. f(x) = 5 − 2x. Find the input x for which f(x) = 11.
Level 2 · Building
4. h(x) = x² − 2x is defined only for the whole numbers x = −1, 0, 1, 2 and 3. How many different values are in its range?
5. A taxi in Accra charges 12 cedi plus 4 cedi for each kilometre, so the fare is C(d) = 12 + 4d for a trip of d km. A trip costs 50 cedi. How many kilometres is it?
Level 3 · Stretch
6. m(x) = √(3x − 12) is only defined when 3x − 12 is not negative. What is the smallest whole-number input?
7. f(x) = 6x − x². There are two inputs for which f(x) = 0. One is x = 0. What is the other?
Something to think about
A function may send two different inputs to the same output, but never one input to two outputs. Is there a rule where every output comes from exactly one input? Try y = 2x + 1 and y = x².
One way to think about it
y = 2x + 1 works: each output comes from just one input, so you can undo it. y = x² does not: 3 and −3 both give 9, so from the output 9 you cannot tell which input was used. Restrict the inputs to x ≥ 0 and the squares become one-to-one again. This is why a function can be undone only when no two inputs share an output.
Exit ticket
8. Which of these is not a function of x?