Functions

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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.

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 $(e.g., x \ne 0$ for 1/x) and square roots of negatives $(e.g., x \ge 0$ $for \sqrt{x})$ .
Range depends on domain; sketch the graph to help find the range.
For a linear function $f(x)=mx+c$ on $a \le x \le b$ , 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 $f^{2}(x)$ 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 $f(x)=x^{2}$ , domain all real numbers, range $f(x) \ge 0$ .
For $f(x)=\frac{1}{x}$ , domain $x\ne 0$ , range $f(x)\ne 0$ .
For $f(x)=\sqrt{x}$ , domain $x\ge 0$ , range $f(x)\ge 0$ .
Sketching helps visualise domain and range.

Composite Functions Algebraically

To find gf(x) algebraically, substitute f(x) into g.
Example: $f(x)=2x-1, g(x)=x^{2}$ → $gf(x) = (2x-1)^{2}$ .
Simplify the resulting expression.
Be careful with order: $gf(x) \ne fg(x)$ 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.

Function Machine

Mapping Diagram

Domain and Range on Graph

Inverse Function Cancellation

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 $f(x) = \frac{1}{x}$ ?
Easy
- AAll real numbers
- B $x \ne 0$
- C $x > 0$
- D $x \ge 0$
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 $f(x) = 2x^{2}$ and $g(x) = 4x^{3}$ , find fg(1).
Medium
- A8
- B32
- C16
- D4
7.If $f(x) = 7x - 4$ and $g(x) = 2x/(x-3), x \ne 3$ , 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}

Summary

## 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)).

Key terms

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).

Slides

What is a Function?

A function is a mathematical 'machine' that takes an input and produces an output.
It applies one or more operations, e.g. $f(x) = 2x + 1$ means 'double and add 1'.
Notation: $f(x) =$ ... where x is the input and f(x) is the output.

For the function $f(x) = 3x - 5$ , what is f(2)?

1
6
-1
11

Domain and Range

Domain: set of all possible inputs (x-values).
Range: set of all possible outputs (f(x)-values).
Domain restrictions: avoid division by zero and square roots of negatives.
Example: $f(x) = \frac{1}{x}$ has domain $x \ne$ 0; $f(x) = \sqrt{x}$ has domain $x \ge 0$ .

What is the domain of $f(x) = 1/(x-3)$ ?

All real numbers
$x \ne 3$
$x > 3$
$x \ge 3$

Finding Range from Domain

To find the range, substitute the domain endpoints into the function.
Consider the shape of the graph (e.g. linear, quadratic).
Example: $f(x) = 10 - x$ with domain $2 < x \le 4$ gives range $6 \le f(x) < 8$ .

For $f(x) = 3x - 5$ with domain {-3, 0, 2}, the range is {____, ____, ____}.

Composite Functions

A composite function applies one function to the output of another.
Notation: gf(x) means do f first, then g (right to left).
Example: $f(x)=2x+1, g(x)=\frac{1}{x}$ . $gf(2) = g(f(2)) = g(5) = \frac{1}{5}$ .

If $f(x)=2x-1$ and $g(x)=x^{2}$ , find fg(3).

Composite Functions Algebraically

To find gf(x) algebraically, substitute f(x) into g.
Example: $f(x)=2x-1, g(x)=x^{2}$ . $gf(x)=g(f(x))=(2x-1)^{2}$ .
Be careful with order: fg(x) means g first, then f.

If $f(x)=3x+2$ and $g(x)=x^{2}+1$ , then $fg(x) = 3(____)+2 = 3x^{2}+5$ .

Inverse Functions

An inverse function reverses the original function's operations.
Notation: f⁻¹(x) is the inverse of f(x).
If $f(a)=b$ , then f⁻¹(b)=a.

If $f(x)=2x+1$ , what is f⁻¹(7)?

Finding Inverse Algebraically

Step 1: Write $y = f(x)$ .
Step 2: Swap x and y.
Step 3: Rearrange to make y the subject.
Step 4: Replace y with f⁻¹(x).

For $f(x)=3x-5$ , the inverse is f⁻¹(x) $= (x+____)/3$ .

Domain and Range of Inverse

Domain of f⁻¹ = Range of f.
Range of f⁻¹ = Domain of f.
Always write domains in terms of x, ranges in terms of f⁻¹(x).

Which statements are true?

The domain of f⁻¹ is the same as the range of f.
The range of f⁻¹ is the same as the domain of f.
f⁻¹(x) $= \frac{1}{f}(x)$ .
ff⁻¹(x) $= x$ .

Using Inverse to Solve Equations

If f⁻¹(x)=a, then $x=f(a)$ because ff⁻¹ cancels.
Useful when finding the inverse algebraically is difficult.
Example: f(x)=2ˣ, solve f⁻¹(x)=5 ⇒ $x=f(5)=32$ .

If $f(x)=x^{3}$ , solve f⁻¹(x)=2.

Summary

Functions map inputs to outputs.
Domain: allowed inputs; Range: possible outputs.
Composite functions: apply one after another (right to left).
Inverse functions: reverse the mapping (swap x and y).

Given $f(x)=4x-1$ with domain {0,1,2}, what is the range?

{-1,3,7}
{0,1,2}
{4,5,6}
{1,5,9}

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