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Odd or Even Functions

3.1 Introduction to Functional Relations

Function Notation and Identities

  • Many of the properties which have been investigated for the functions introduced in the previous chapters may be expressed using function notation.
  • Below includes a few examples:

\ln x+\ln y=\ln (x y) becomes f(x)+f(y)=f(x y) where f(k)=\ln k

\ln x-\ln y=\ln \left(\frac{x}{y}\right) becomes f(x)-f(y)=f\left(\frac{x}{y}\right) where f(k)=\ln k

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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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