Chapter 1 of 8

All 8 chapters
  1. 1Function notation, readunderstood
  2. 2Composite and inverse functions, readunderstood
  3. 3Cubic, reciprocal and circle graphs, readunderstood
  4. 4Exponential graphs, readunderstood
  5. 5Graphs of sin, cos and tan, readunderstood
  6. 6Moving graphs, readunderstood
  7. 7Gradients and areas under graphs, readunderstood
  8. 8Solving with graphs and trials, readunderstood

M4.10.1 Chapter 1

Function notation

My book says f(x) = 2x − 1. Is f a number that multiplies x?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find f(3) when f(x) = 2x − 1
  • Solve f(x) = 7 for the same function

A function is a rule. You give it a number, and it gives back one number. The number you give is the input. The number you get back is the output.

A row of boxes joined by arrows. The first box is x. An arrow named × 2 leads to the box 2x. An arrow named − 1 leads to the box f(x).x× 22x− 1f(x)
The rule f: double the input, then take away 1. The output is called f(x).

We write this rule as f(x) = 2x − 1. Read it as "f of x equals 2 x minus 1". The letter f is the name of the rule. The letter x stands for any input. And f(x) is the output.

The brackets in f(x) are not a multiplication. Do not read f(x) as f times x. They show which number is the input.

Two columns joined by arrows. The column x holds 1, 2, 3 and 4. The column f(x) holds 1, 3, 5 and 7. Each number in x has one arrow to the number in f(x) beside it: 1 to 1, 2 to 3, 3 to 5 and 4 to 7.xf(x)12341357
f(x) = 2x − 1 as arrows. Each input has exactly one output.

In the picture the input 3 goes to the output 5. We write that as f(3) = 5. Notice that each input has one arrow only. That is what makes a rule a function.

Draw it first

From a rule to a graph

  1. A graph with the x axis x and the y axis y. Step 1 plots the points (0, −1), (1, 1), (2, 3), (3, 5) and (4, 7) as dots. Step 2 joins them with a straight line. Step 3 marks the point A at (3, 5) with dashed lines to both axes.12345−2−1123456780xy

    1 Work out f(x) for several inputs, and plot each pair (x, f(x)) as a point.

  2. A graph with the x axis x and the y axis y. Step 1 plots the points (0, −1), (1, 1), (2, 3), (3, 5) and (4, 7) as dots. Step 2 joins them with a straight line. Step 3 marks the point A at (3, 5) with dashed lines to both axes.12345−2−1123456780xy

    2 The points lie on a straight line. Join them with a ruler.

  3. A graph with the x axis x and the y axis y. Step 1 plots the points (0, −1), (1, 1), (2, 3), (3, 5) and (4, 7) as dots. Step 2 joins them with a straight line. Step 3 marks the point A at (3, 5) with dashed lines to both axes.A12345−2−1123456780xy

    3 To read f(3), go up from x = 3 to the line, then across to the y axis. You land on 5.

The graph of a function is the set of all points (x, f(x)). That is why we often write y = f(x). The y value of the graph is the output.

Worked example

f(x) = 2x − 1. Find f(3).

  1. Write the rule: f(x) = 2x − 1.
  2. Put 3 in place of every x: f(3) = 2 × 3 − 1.
  3. Multiply first: 2 × 3 = 6.
  4. Then take away 1: 6 − 1 = 5.

Answer: f(3) = 5.

Now a rule with a square. Take care with a negative input. Put the input in a bracket, then square the whole bracket.

Worked example

f(x) = x² − 3. Find f(−2).

  1. Write the rule: f(x) = x² − 3.
  2. Put −2 in a bracket in place of x: f(−2) = (−2)² − 3.
  3. Square the bracket: (−2) × (−2) = 4.
  4. Take away 3: 4 − 3 = 1.

Answer: f(−2) = 1.

Sometimes you are told the output and must find the input. Then you write f(x) equal to that number, and solve the equation. This means undoing the rule, step by step, in the opposite order.

Two rows of boxes joined by arrows. The row named Do the rule goes x, then 2x, then 7, with arrows × 2 and − 1. The row named Undo it goes 7, then 2x, then x, with arrows + 1 and ÷ 2.Do the rulex× 22x− 17Undo it7+ 12x÷ 2x
To find x, run the machine backwards: add 1, then divide by 2.

Worked example

f(x) = 2x − 1. Solve f(x) = 7.

  1. Write f(x) = 7 as an equation: 2x − 1 = 7.
  2. Add 1 to both sides: 2x = 8.
  3. Divide both sides by 2: x = 4.
  4. Check by putting 4 in the rule: f(4) = 2 × 4 − 1 = 7.
  5. The output is 7, as wanted. So the input is 4.

Answer: x = 4.

See it three ways

With things

Amina doubles her number of beans, then gives one bean away. She starts with 3 beans.

As a picture

A row of boxes joined by arrows. The first box is 3. An arrow named × 2 leads to the box 6. An arrow named − 1 leads to the box 5.3× 26− 15

In words

Double the input, then take away 1. The input 3 gives the output 5.

In symbols

f(x) = 2x − 1, so f(3) = 2 × 3 − 1 = 5.

Try it

A table for f(x) = 2x − 1

You need: paper, a pencil.

  1. Write the numbers 0, 1, 2, 3 and 4 in a row. These are the inputs.
  2. Under each one, work out 2x − 1. These are the outputs f(x).
  3. Look along the outputs. By how much does each one differ from the one before?

What you will see: The outputs go up by 2 each time, the number in front of x. When the input is 0, the output is −1, the number you take away.

If you cannot try it, this is what happens.
A table with the heads x and f(x). Rows: 0 and −1; 1 and 1; 2 and 3; 3 and 5; 4 and 7.xf(x)0−111233547

A wrong idea many people have

f(3) means f times 3.

Why it feels right: In algebra, letters next to each other multiply, as in 3x. So f(3) looks like a product.

The test that breaks it: Take f(x) = 2x − 1. Then f(1) = 1. If f were a number, f(1) = f × 1 would make f equal to 1. Then f(2) = f × 2 would be 2. But f(2) = 2 × 2 − 1 = 3.

What is true: f is the name of a rule, not a number. The brackets hold the input. f(3) is the output when the input is 3.

A wrong idea many people have

f(x) = 7 means find f(7).

Why it feels right: The number 7 is in the question. The expression f(7) also has a 7 in it, so they seem the same job.

The test that breaks it: For f(x) = 2x − 1, f(7) = 13. But f(4) = 7. The equation f(x) = 7 asks for the input 4.

What is true: In f(7) the 7 is the input. In f(x) = 7 the 7 is the output. Solve the equation to find the input.

Key words

function
a rule that gives exactly one output for each input.
input
the number you put into the function, shown as x in f(x).
output
the number that comes out of the function, written f(x).

Back to your guess

The answer is C: No. f is the name of a rule, and f(3) is what comes out when 3 goes in.

The letter f is a name, like the name of a machine. The brackets do not mean times. They hold the input. So f(3) means: put 3 into the rule f and see what comes out. For f(x) = 2x − 1, double 3, take away 1, and the output is 5.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. f(x) = 3x + 2. Find f(4).

    Tap your answer.

    1. Not this one. You added 3 + 4 + 2. The 3x means 3 times x, so the first step is 3 × 4 = 12. Go to M4.10.1. Try another.

    2. Not this one. You wrote the 3 and the 4 side by side, as 34. But 3x means 3 times x, so it is 3 × 4 = 12. Go to M3.8.2. Try another.

    3. Yes, that is right. Put 4 in place of x: f(4) = 3 × 4 + 2. Multiply first: 12. Then add 2. The output is 14.

    See the answer

    14

    Put 4 in place of x: f(4) = 3 × 4 + 2. Multiply first: 12. Then add 2. The output is 14.

    • If you chose A: You added 3 + 4 + 2. The 3x means 3 times x, so the first step is 3 × 4 = 12. Go to M4.10.1.
    • If you chose B: You wrote the 3 and the 4 side by side, as 34. But 3x means 3 times x, so it is 3 × 4 = 12. Go to M3.8.2.

  2. f(x) = x² + 1. Find f(−3).

    Tap your answer.

    1. Yes, that is right. Put −3 in a bracket: f(−3) = (−3)² + 1. The square is (−3) × (−3) = 9. Then 9 + 1 = 10.

    2. Not this one. You took (−3)² as −9. But (−3) × (−3) is +9, because two negative numbers multiply to a positive one. So f(−3) = 9 + 1. Go to M4.10.1. Try another.

    3. Not this one. You read x² as x times 2, which is −6. But x² means x times x, and (−3) × (−3) = 9. Go to M4.10.1. Try another.

    See the answer

    10

    Put −3 in a bracket: f(−3) = (−3)² + 1. The square is (−3) × (−3) = 9. Then 9 + 1 = 10.

    • If you chose B: You took (−3)² as −9. But (−3) × (−3) is +9, because two negative numbers multiply to a positive one. So f(−3) = 9 + 1. Go to M4.10.1.
    • If you chose C: You read x² as x times 2, which is −6. But x² means x times x, and (−3) × (−3) = 9. Go to M4.10.1.

  3. The picture shows a function g as arrows. What is g(4)?

    Two columns joined by arrows. The column x holds 1, 2, 3, 4 and 5. The column g(x) holds 5, 7, 9, 11 and 13. Each number in x has one arrow to the number in g(x) at the same height.xg(x)123455791113
    Tap your answer.

    1. Not this one. The number 4 is the input. The answer is the number at the end of the arrow that starts at 4. Go to M4.10.1. Try another.

    2. Not this one. The number 9 is the output for the input 3, one place higher up. Follow the arrow that starts at 4. Go to M4.10.1. Try another.

    3. Not this one. You doubled 4. But the arrow from 4 ends at 11, so the rule does more than double. Go to M4.10.1. Try another.

    4. Yes, that is right. g(4) is the output for the input 4. Follow the arrow from 4: it ends at 11.

    See the answer

    11

    g(4) is the output for the input 4. Follow the arrow from 4: it ends at 11.

    • If you chose A: The number 4 is the input. The answer is the number at the end of the arrow that starts at 4. Go to M4.10.1.
    • If you chose B: The number 9 is the output for the input 3, one place higher up. Follow the arrow that starts at 4. Go to M4.10.1.
    • If you chose C: You doubled 4. But the arrow from 4 ends at 11, so the rule does more than double. Go to M4.10.1.

  4. f(x) = 4x − 3. Solve f(x) = 17.

    Tap your answer.

    1. Not this one. That is f(17), with 17 as the input. Here 17 is the output, and you want the input. Go to M4.10.1. Try another.

    2. Yes, that is right. Write 4x − 3 = 17. Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    3. Not this one. You took away 3 from 17. To undo "take away 3" you must add 3: 4x = 17 + 3 = 20. Go to M3.8.6. Try another.

    See the answer

    5

    Write 4x − 3 = 17. Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    • If you chose A: That is f(17), with 17 as the input. Here 17 is the output, and you want the input. Go to M4.10.1.
    • If you chose C: You took away 3 from 17. To undo "take away 3" you must add 3: 4x = 17 + 3 = 20. Go to M3.8.6.

  5. For f(x) = 2x + 2, we find f(5) = 12. What does this tell you?

    Tap your answer.

    1. Not this one. The f is not a number to multiply by. It is the name of the rule. Go to M4.10.1. Try another.

    2. Not this one. The number in the brackets is the input. In f(5) = 12, the input is 5 and the output is 12. Go to M4.10.1. Try another.

    3. Not this one. The brackets do not mean add. They hold the input of the rule. Go to M4.10.1. Try another.

    4. Yes, that is right. In f(5) = 12 the 5 is in the brackets, so it is the input. The 12 is the output. Check: 2 × 5 + 2 = 12.

    See the answer

    The input 5 gives the output 12.

    In f(5) = 12 the 5 is in the brackets, so it is the input. The 12 is the output. Check: 2 × 5 + 2 = 12.

    • If you chose A: The f is not a number to multiply by. It is the name of the rule. Go to M4.10.1.
    • If you chose B: The number in the brackets is the input. In f(5) = 12, the input is 5 and the output is 12. Go to M4.10.1.
    • If you chose C: The brackets do not mean add. They hold the input of the rule. Go to M4.10.1.

  6. The graph shows y = f(x) for f(x) = x − 2. For which x is f(x) = 1?

    A graph with the x axis x and the y axis y. A straight line rises through the points (0, −2) and (2, 0). A dashed horizontal line at height 1 meets the straight line at a point named P.P2345−3−2−11230xy
    Tap your answer.

    1. Yes, that is right. The dashed line is at height 1, so the output is 1. It meets the graph at P. Go straight down from P to the x axis: you land on 3. Check: 3 − 2 = 1.

    2. Not this one. The number 1 is the height, the output. The question asks for the x value of the point P, which is the input. Go to M4.10.1. Try another.

    3. Not this one. The line crosses the x axis at x = 2, where the output is 0. The output you want is 1, the height of the dashed line. Go to M4.10.1. Try another.

    See the answer

    3

    The dashed line is at height 1, so the output is 1. It meets the graph at P. Go straight down from P to the x axis: you land on 3. Check: 3 − 2 = 1.

    • If you chose B: The number 1 is the height, the output. The question asks for the x value of the point P, which is the input. Go to M4.10.1.
    • If you chose C: The line crosses the x axis at x = 2, where the output is 0. The output you want is 1, the height of the dashed line. Go to M4.10.1.

  7. f(x) = 5 − 3x. Find f(−2).

    See the answer

    11

    Put −2 in place of x: f(−2) = 5 − 3 × (−2). Now 3 × (−2) = −6, and taking away −6 is adding 6. So f(−2) = 5 + 6 = 11.

  8. f(x) = 3x + 4. Solve f(x) = 19.

    See the answer

    x = 5

    Write 3x + 4 = 19. Take away 4 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 4 = 19.

Keep it

Copy this into your exercise book.

A function is a rule. Each input has one output. f(3) means put 3 in. f(x) = 7 means find the x that gives 7. Undo the rule backwards.

Draw: A machine with the arrows x, then × 2, then − 1, then f(x), and a second row going back.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (956 KB). It opens on any phone.