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
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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.

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1.7 Additional Graph Types


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  • 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].
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1.6 Trigonometric Functions

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).

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1.4 Logarithmic Functions

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’.


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1.3 Exponential Functions [Free]

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.


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

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1.2 Power Functions

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

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1.1 Introduction to Graph Sketching

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.

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