Functions

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Lesson notes

Introduction to Functions

A function is a mathematical 'machine' that takes an input and produces an output.
Function notation: f(x) = ... means the function f with input x.
Common letters for functions: f, g, h, j.
To evaluate f(a), substitute a for x in the expression.
If f(x) = output is known, solve f(x) = value to find the input x.

Mapping diagram for a function

Domain & Range

The domain is the set of all possible inputs (x-values).
The range is the set of all possible outputs (f(x)-values).
Domain restrictions: avoid division by zero (e.g., x ≠ 0 for 1/x) and square roots of negatives (e.g., x ≥ 0 for √x).
Range depends on domain; sketch the graph to help find the range.
For a linear function f(x)=mx+c on a ≤ x ≤ b, the range is between f(a) and f(b) (order depends on gradient).

Composite Functions

A composite function applies one function to the output of another.
Notation: gf(x) means do f first, then g: gf(x) = g(f(x)).
fg(x) means do g first, then f: fg(x) = f(g(x)).
ff(x) or f²(x) means apply f twice.
To evaluate numerically: work from the inside out.

Inverse Functions

An inverse function reverses the original function; notation: f⁻¹(x).
If f(a)=b, then f⁻¹(b)=a.
To find f⁻¹(x) algebraically: write y = f(x), swap x and y, then solve for y.
The composite of a function and its inverse cancels: ff⁻¹(x) = f⁻¹f(x) = x.
Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.

Finding Inverse Functions

Step 1: Write y = f(x).
Step 2: Swap x and y to get x = f(y).
Step 3: Rearrange to make y the subject.
Step 4: Replace y with f⁻¹(x).
Example: f(x)=2x+1 → f⁻¹(x) = (x-1)/2.

Domain and Range from Graphs

The graph of y = f(x) shows domain on x-axis and range on y-axis.
For f(x)=x², domain all real numbers, range f(x) ≥ 0.
For f(x)=1/x, domain x≠0, range f(x)≠0.
For f(x)=√x, domain x≥0, range f(x)≥0.
Sketching helps visualise domain and range.

Graph of y = g(x) = 3x^2, domain x >= 0

Composite Functions Algebraically

To find gf(x) algebraically, substitute f(x) into g.
Example: f(x)=2x-1, g(x)=x² → gf(x) = (2x-1)².
Simplify the resulting expression.
Be careful with order: gf(x) ≠ fg(x) generally.

Inverse Functions and Solving Equations

If f⁻¹(x)=c, then x = f(c) using cancellation.
Example: f(x)=2x, solve f⁻¹(x)=5 → x = f(5)=10.
This avoids finding f⁻¹ explicitly.
Useful when inverse is difficult to find algebraically.

Slides

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Practice questions

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1.A function is defined as f(x) = 3x - 5. What is the value of f(2)?
Easy
- A1
- B6
- C-1
- D11
2.The domain of a function f(x) is the set of all possible inputs. Which of the following is the correct domain for f(x) = 1/x?
Easy
- AAll real numbers
- Bx ≠ 0
- Cx > 0
- Dx ≥ 0
3.For the function f(x) = 2x + 1, what is the output when the input is -3?
Easy
- A-5
- B5
- C-7
- D7
4.Given f(x) = 7 - 2x, find f(-3).
Easy
- A13
- B1
- C-13
- D-1
5.The function f is defined by f(x) = 3x - 5. The domain of f is {-3, 0, 2}. What is the range of f?
Medium
- A{-14, -5, 1}
- B{-4, -5, 1}
- C{-14, -5, 11}
- D{ -4, -5, 11}
6.Given f(x) = 2x² and g(x) = 4x³, find fg(1).
Medium
- A8
- B32
- C16
- D4
7.If f(x) = 7x - 4 and g(x) = 2x/(x-3), x ≠ 3, find fg(4).
Medium
- A52
- B24
- C60
- D38
8.The function f is defined by f(x) = 2x - 7. The domain of f is {-2, 0, 5}. What is the range of f?
Medium
- A{-11, -7, 3}
- B{-11, -7, 17}
- C{-3, -7, 3}
- D{-11, 7, 3}

More questions

9.Given f(x) = 3x + 2 and g(x) = x² + 1, find fg(1).
Medium
- A8
- B14
- C5
- D11
10.Given f(x) = 3x + 2 and g(x) = x² + 1, find gf(x) in the form ax² + bx + c.
Hard
- A9x² + 12x + 5
- B3x² + 5
- C9x² + 12x + 4
- D3x² + 12x + 5
11.Find the inverse function f^-1(x) for f(x) = 4x - 1.
Medium
- Af^-1(x) = (x+1)/4
- Bf^-1(x) = (x-1)/4
- Cf^-1(x) = 4x + 1
- Df^-1(x) = x/4 + 1
12.Given h(x) = 4/x, find x when h^-1(x) = 2.
Medium
- A2
- B8
- C1/2
- D4
13.The function f is defined by f(x) = 3x² + a, where a is an integer. Given that f(-2) = 19, find the value of a.
Medium
- A7
- B1
- C31
- D13
14.Given g(x) = 2x + 3 and h(x) = 2/x, x ≠ 0, find x when gh(x) = 7.
Medium
- A1
- B0.5
- C2
- D2.5
15.Given f(x) = 7x - 4, find the value of x when f(x) + 2 = -11.
Medium
- A-1
- B-13/7
- C-9/7
- D-2
16.A function is defined as f(x) = 5 - 2x. Find f(3).
Easy
- A-1
- B1
- C11
- D-11
17.Given f(x) = 3x + 4, what is the value of f(-2)?
Easy
- A-2
- B2
- C10
- D-10
18.The function f is defined by f(x) = 2x - 1. Find the value of x for which f(x) = 9.
Easy
- A4
- B5
- C3
- D6
19.Given f(x) = x² + 1 and g(x) = 2x - 3, find fg(2).
Medium
- A2
- B6
- C10
- D26
20.Given f(x) = 3x - 2, find the inverse function f^-1(x).
Medium
- Af^-1(x) = (x+2)/3
- Bf^-1(x) = (x-2)/3
- Cf^-1(x) = (x+3)/2
- Df^-1(x) = 3x+2
21.The function f is defined by f(x) = 7 - 3x. The domain of f is {-1, 0, 4}. Find the range of f.
Medium
- A{10, 7, -5}
- B{4, 7, -5}
- C{10, 7, 5}
- D{4, 7, 5}
22.Given f(x) = 2x + 5 and g(x) = x², find gf(3).
Medium
- A121
- B11
- C31
- D19
23.Given f(x) = 4x - 1 and g(x) = x², find fg(x) in its simplest form.
Medium
- A4x² - 1
- B16x² - 1
- C(4x-1)²
- D4x² - 4x + 1
24.Given f(x) = 3x + 2 and g(x) = 2/x, x ≠ 0, find the value of x such that fg(x) = 8.
Medium
- A1/2
- B2
- C1
- D3/2
25.Given f(x) = 2x - 7, find the value of x when f(x) + 3 = -2.
Easy
- A1
- B6
- C-1
- D3
26.The function h is defined by h(x) = 3/x, x ≠ 0. Find x when h^-1(x) = 6.
Medium
- A2
- B18
- C1/2
- D1/18
27.Given f(x) = 5x - 2 and g(x) = 3x + 1, find the value of x for which f(x) = g(2).
Medium
- A3
- B1
- C2
- D4
28.Given f(x) = 2x² and g(x) = 3x, find gf(2).
Medium
- A24
- B12
- C144
- D36
29.The function f is defined by f(x) = 3x² + a, where a is an integer. Given that f(2) = 17, find the value of a.
Easy
- A5
- B-5
- C11
- D-11
30.Given g(x) = 2x + 3, find x when gg(x) = 7.
Medium
- A-0.5
- B0.5
- C1
- D-1
31.Given g(x) = 2x + 3 and h(x) = 2/x, find x when h^-1(x) = 5.
Medium
- A10
- B0.4
- C2.5
- D0.5
32.A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers. What are the numbers being put into the function called?
Easy
- AInputs
- BOutputs
- CDomains
- DRanges
33.For the function f(x) = 1 / x, which value of x must be excluded from the domain?
Easy
- Ax = 0
- Bx = 1
- Cx = -1
- Dx = 2
34.Given f(x) = 3x + 4, what is the value of f(0)?
Easy
- A4
- B0
- C3
- D7
35.The function f is defined by f(x) = 2x - 1. The domain of f is {-1, 0, 3}. What is the range of f?
Medium
- A{-3, -1, 5}
- B{-3, -1, 7}
- C{-1, 1, 5}
- D{-2, -1, 5}
36.Given f(x) = 2x² and g(x) = 3x, find fg(2).
Medium
- A72
- B24
- C36
- D48
37.Which of the following is the correct definition of a function?
Easy
- AA relation where each input has exactly one output
- BA relation where each output has exactly one input
- CA relation where inputs and outputs are always numbers
- DA relation where inputs are multiplied by outputs
38.The function f(x) = 1/x is defined for all real values of x.
Easy
True or false?
39.Given f(x) = 2x + 1 and g(x) = x², what is fg(3)?
Medium
- A19
- B37
- C10
- D16
40.Which of the following are correct statements about the function f(x) = x² with domain all real numbers? (Select all that apply)
Hard
- AThe range is f(x) ≥ 0
- BThe function is one-to-one
- Cf(-3) = 9
- DThe inverse function is f^-1(x) = √(x)
- Ef(0) = 0
41.Match each function with its correct inverse.
Medium
- f(x) = 3x + 2
- g(x) = x - 5
- h(x) = 2/x, x ≠ 0
- f^-1(x) = (x-2)/3
- g^-1(x) = x + 5
- h^-1(x) = 2/x, x ≠ 0
42.Arrange the following steps in the correct order to find the inverse of f(x) = 5x - 3.
Medium
- Write y = 5x - 3
- Swap x and y: x = 5y - 3
- Solve for y: y = (x+3)/5
- Write f^-1(x) = (x+3)/5
43.If f(x) = 2x and g(x) = x + 3, then fg(x) = gf(x).
Medium
True or false?
44.The function f is defined by f(x) = \frac{3}{x-2}, x \neq 2. What is the value of f(5)?
Easy
- A1
- B3
- C\frac{3}{7}
- D\frac{3}{3}
45.Write down the mathematical notation for the inverse function of f(x).
Easy
46.If f(x) = 4x - 3, find f(2).
Easy
47.Complete the sentence: The set of all possible inputs of a function is called the ___.
Easy
The set of all possible inputs of a function is called the ____.
48.Given f(x) = 2x² - 3 and g(x) = 4x + 1, find the positive value of x such that fg(x) = 5.
Medium
49.What is the range of f(x) = 3 - x² with domain all real numbers?
Medium
50.Given f(x) = 3x² - 2x + 1, find f(7).
Easy
51.Given f(x) = 4x - 1, find the value of x when f(x) + 2 = -11.
Medium
52.Given f(x) = 5 - 2x and g(x) = x² + 1, find fg(3).
Medium
53.Complete the sentence.
Easy
The set of all possible outputs of a function is called the ____.
54.Given f(x) = 3x - 2, find f^-1(x).
Medium
55.The function f is defined by f(x) = 2x² - 5. The domain of f is {-2, 0, 3}. What is the smallest value in the range of f?
Medium
56.Given f(x) = 2x + 3 and g(x) = \frac{4}{x}, x \neq 0, find the value of x such that fg(x) = 7.
Medium
57.Explain why the function f(x) = x², with domain all real numbers, does not have an inverse function unless the domain is restricted.
Hard
58.If f(x) = 4x - 3, find f(2).
Easy
- A4
- B6
- C5
- D7
59.Given f(x) = 2x² - 3 and g(x) = 4x + 1, find the positive value of x such that fg(x) = 5.
Medium
- A2.5
- B0.25
- C-0.75
- D1.5
60.Given f(x) = 3x² - 2x + 1, find f(7).
Easy
- A147
- B134
- C160
- D121
61.Given f(x) = 4x - 1, find the value of x when f(x) + 2 = -11.
Medium
- A-2
- B-4
- C-1
- D-3
62.Given f(x) = 5 - 2x and g(x) = x² + 1, find fg(3).
Medium
- A-14
- B-16
- C-13
- D-15
63.The function f is defined by f(x) = 2x² - 5. The domain of f is {-2, 0, 3}. What is the smallest value in the range of f?
Medium
- A-6
- B-4
- C-5
- D-3
64.Given f(x) = 2x + 3 and g(x) = \frac{4}{x}, x \neq 0, find the value of x such that fg(x) = 7.
Medium
- A3
- B4
- C2
- D1

Revision notes

## Functions - IGCSE Maths Extended ### Introduction - A function maps inputs to outputs via operations. Notation: f(x) = ... - To evaluate, substitute the input. E.g., f(x)=2x+1, f(3)=7. - To find input given output, solve f(x)=output. ### Domain & Range - **Domain**: set of allowed inputs (x-values). Exclude values causing division by zero or square roots of negatives. - **Range**: set of outputs (f(x)-values). Depends on domain. - Use graphs to visualise range. ### Composite Functions - gf(x) means apply f first, then g: g(f(x)). - fg(x) means apply g first, then f. - ff(x) = f(f(x)). - To evaluate numerically, work inside out. ### Inverse Functions - f⁻¹(x) reverses f. If f(a)=b then f⁻¹(b)=a. - To find algebraically: write y=f(x), swap x and y, solve for y, replace y with f⁻¹(x). - Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f. - ff⁻¹(x) = f⁻¹f(x) = x (cancellation). ### Key Formulas - f(x)=3x-5, domain {-3,0,2} → range {-14,-5,1}. - f(x)=7-2x, f⁻¹(x) = (7-x)/2. - f(x)=3/(x+2), x≠-2. - g(x)=8x-5, g⁻¹(x) = (x+5)/8. - Composite: fg(x) = f(g(x)).

Function: A rule that maps each input to exactly one output.
Domain: The set of all possible inputs (x-values) for a function.
Range: The set of all possible outputs (f(x)-values) of a function.
Composite function: A function applied to the output of another function; e.g., gf(x) = g(f(x)).
Inverse function: A function that reverses the effect of the original function; denoted f⁻¹(x).

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