Domain

4.1 Composite Functions

Composite Functions

A composite function is different from sum or product of functions; it is like a function acting on the result of another. If you picture functions as machines (with $x$ as the input, and $f(x)$ as the output), it is something like the diagram shown.

Note: Here, $f(x)=3 x+2,\ g(x)=x^{2}$ , and $x=3$ .

In comparison, $(f+g)(x)$ is more like tow machines working separately with the same input $x$ , and their two, separate outputs are placed together and packaged.

Just like how the processes of manufacturing has an order, so does composite functions. It is important which functions comes first, and which comes next. Using the example above where $f$ is the first function), we write the composition of $g$ with $f$ (say we call it as $h$ ) as $h=g \circ f$ (read as ‘composition of $f$ followed by $g$ ‘). The rule of $h(x)$ is given by

1.2 Describing a Set and Function

Describing a Set

It is not always possible to list all the elements in a set (e.g. N, Z, Q, R as above). There is a way of describing sets that is useful for infinite sets (i.e. sets with an infinite number of elements).

The set of all $x$ such that ____ is denoted as $\left\{x:_{-----}\right\}$ . For example:

i) $\{x:0<,\ x<1\}$ is the set of real numbers strictly between 0 and 1.

ii) $\{x:x\leq3\}$ is the set of real numbers smaller or equal to 3.

iii) $\{x: x>0, x \in Q\}$ is the set of all positive rational numbers.

Plotting Logarithmic Functions

Example

Say $y=log_{e}{x}$ . Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2).

Generally, the steps to plot a graph are:

1. Identify the domain and range of the function.

Plotting Exponential Functions

Example

Say $y=2^{x}$ . Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2). You should try yourself for $y=0.5^{x}$ .

Generally, the steps to plot a graph are:

Circles

A picture containing chart Description automatically generated

The equation of a ‘standard’ circle is $x^{2}+y^{2}=r^{2}$ .
This circle has a radius of $r$ , and its centre is the origin $(0,\ 0)$ .

E.g. The right shows the graph of the circle with equation $x^{2}+y^{2}=4$ .

Note: Be careful here that the radius is 2, and not 4 as it should be $r^{2}=4,\ r=2$ .

By definition $r>0$ as it signifies the radius of the circle. However, if we observe the equation, $r$ can be negative too (the interpretation of the negative sign will involve a concept called ‘vectors’).
As we can see, $x$ and $y$ both have a range of $[-r,\ r]$ .

Graphs of Sine and Cosine Functions

The graph of where $-\pi<x<3\pi$ are plotted below. Do note that it extends beyond the drawn range.

There are a few observations we can make, and we can tie it back to what we have already learnt:

i) The graph repeats itself after an interval of $2\pi$ units. We say that the graph has a period of $2\pi$ , and hence is called a periodic function. Previously, we have learnt that $\sin(x+2\pi)$ .

The Logarithmic Function

The logarithmic function with base is defined as follows:

$a^{x} = b$ is equivalent to $log_{a}{b}=x$ , where $a \in R^{+}\setminus \left \{ 1 \right \}$ .

Note: ‘ $a \in R^{+}\setminus \left \{ 1 \right \}$ ‘ means that could be any positive number, excluding 1.

The expression $log_{a}{y}=x$ is read as: ‘the logarithm of $y$ to the base $a$ is equal to $x$ ’.

Example

The Exponential Function

The exponential function is defined as follows:

$y=a^x$ , where $a \in R^+\backslash {1}$ .

Note: ‘ $a \in R^+\backslash {1}$ ’ means that could be any positive number, excluding 1.

Example

$9=3^{2}=3 \times 3$ is equivalent to $log_{3}9=2$ .

$8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}$ .

Graph & Properties

Introduction to Power Functions

Power functions are functions with the form $f(x)=x^{n}$ , where $n$ is a rational number.
Some examples of power functions are:

$f(x)=x^{2},\ f(x)=x^{4},\ f(x)=x^{\frac{1}{4}},\ f(x)=x^{-5},\ f(x)=x^{\frac{1}{3}}$

The expression $x^n$ is read as: ‘ $x$ to the power of $n$ ’.

Drawing Power Functions (for Positive Integers) and its Graph Properties

Introduction to Graph Sketching

In this section, 3 types of functions will be introduced accordingly:

i) Power functions, $y=x^n$ , where is a number, $n\neq 0$ .

This will be further extended into polynomial functions, which has the general form $y=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$ , where $a_i$ are real numbers, $a_n\neq 0$ .

ii) Exponential functions, $y=a^x$ and also logarithm functions, $y=log_ax$ .

iii) Trigonometric functions or circular functions, which includes $y=\sin x,\ y=\cos x,\ y=\tan x$ and more.

Domain

4.1 Composite Functions

Composite Functions

1.2 Describing a Set and Function

Describing a Set

3.7 Transformations of Logarithmic Functions

Plotting Logarithmic Functions

3.6 Transformations of Exponential Functions

Plotting Exponential Functions

1.7 Additional Graph Types

Circles

1.6 Trigonometric Functions

Graphs of Sine and Cosine Functions

1.4 Logarithmic Functions

The Logarithmic Function

1.3 Exponential Functions [Free]

The Exponential Function

1.2 Power Functions

Introduction to Power Functions

Drawing Power Functions (for Positive Integers) and its Graph Properties

1.1 Introduction to Graph Sketching

Introduction to Graph Sketching