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# Functions

## 5.4 Solving Exponential and Logarithmic Equations

#### Fundamental of Solving Exponential and Logarithm Functions

• In solving equations, the most important identity of both exponential and logarithmic functions are that they are one-to-one functions. In other words,

a^{x}=a^{y} or \log_{a}{x}=\log_{a}{y} implies that x=y.

Example

3^{x-1}=81=3^{4} Read More »5.4 Solving Exponential and Logarithmic Equations

## 5.3 Solution of Trigonometric Equations

#### Basic Trigonometric Equations (Sine and Cosine)

• The most basic equations in polynomials would be linear equations, such as ax+b=c (which gives you x=\frac{c-b}{a}). The trigonometric equivalent ones would be \sin t =a or \cos t =b, and you are supposed to solve for t.
• Be careful, as solving trigonometric equations are not as simple. Refer to the examples below.
• For each such equations, unless a restriction on x (or more commonly used in trigonometric, \theta), there will be infinite or no solutions for x. This is because of the periodic and symmetric properties that they have.
Read More »5.3 Solution of Trigonometric Equations

## 5.2 Simultaneous Equations: Polynomials

#### Solving Equations

• Every now and then, you will be challenged with questions to solve .
• Suppose we have y=f(x) and y=g(x) as two separate functions. If f(x)=g(x), it means that the graph of both functions intersect. Note that this tells nothing about the number of possible values for x.

#### Simultaneous Equations: Polynomials

• We will first look at how this type of questions looks like, with polynomials first.
• The steps are rather simple:
Read More »5.2 Simultaneous Equations: Polynomials

## 5.1 Application of Functions

#### Application of Functions

• Functions is a very useful tool in expressing relations, even in our daily life. Below are a few examples.
Read More »5.1 Application of Functions

## 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

## 3.4 Trigonometric Function Laws

Note: These are already covered in the notes 1.5 Trigonometric Functions Background Knowledge, when trigonometric functions are first introduced. Here, it will be simply a reiteration of the formulas.

#### Trigonometric Function Laws

• Sine, cosine, tangent can be understood as follows:

\sin (\theta)=\frac{\text { opposite }}{\text { hypotenuse }}

\cos (\theta)=\frac{\text { adjacent }}{\text { hypotenuse }}

\tan (\theta)=\frac{\text { opposite }}{\text { adjacent }}

and in particular we have

Read More »3.4 Trigonometric Function Laws

## 3.3 Logarithm Laws [Free]

#### The Logarithm Function’s Algebraic Properties

• There are four properties in logarithms to consider:

i) \log_{a}{mn}=\log_{a}{m}=\log_{a}{n}

ii) log_{a}{(\frac{m}{n})}=\log_{a}{m}-\log_{a}{n}

iii) log_{a}{(m^{p})}=p\cdot{\log_{a}{m}}

In particular when p=-1, we have log_{a}{(\frac{1}{m})}=-\log_{a}{m}

Read More »3.3 Logarithm Laws [Free]

## 3.2 Index Laws

#### The Exponential Function’s Algebraic Properties

• There are six properties in exponentials to consider:

i) a^{x} \times a^{y} = a^{x+y}

ii) a^{x} {\div} a^{y} = a^{x-y}

iii) (a^{x})^{y}=a^{xy}

## 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

Read More »3.1 Introduction to Functional Relations

## 2.2 Remainder Theorem, Factor Theorem and the Rational-Root Theorem

#### Remainder Theorem and Factor Theorem

Now we are going to set D(x)=x-a. Therefore, as \operatorname{deg}(R)<\operatorname{deg}(D), R(x) must be a constant, which we denote as R.

Rewriting the equation gives

P(x)=(x-a) Q(x)+R

• Simply set x=a gives P(x)=(a-a) Q(x)+R=0+R=R. Thus, R=P(a).
• Now we generalize D(x) into any linear polynomial, i.e. D(x)=a x+b, where a,\ b are integers. Therefore, now P(x)=(a x+b) Q(x)+R and substitute x=-\frac{b}{a} (to make D(x)=0) gives
Read More »2.2 Remainder Theorem, Factor Theorem and the Rational-Root Theorem