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# Sketching the Graph

## 1.4 Inverse Functions

#### Inverse Functions

• A function maps the domain to the range. An inverse function simply does the opposite.
• However, recall that a function must have the type of relation which is a one-to-one relation. Therefore, if the inverse function (which, yes, is a function) exists, the inverse must have a one-to-one relation too.
• This implies that for an inverse function to exist (might be over a certain subset of the domain), the function must be one-to-one (over that particular subset if applicable).
• We denote the inverse of f as f^{-1}. In particular, if f(x)=y, then f^{-1}(y)=x.
• Therefore, we can deduce that:

## 5.5 Sum and Product of Functions

#### Sum and Product of Functions

• Different functions can be added or multiplied together.
• Say there is two functions f(x) and g(x). We write their sum as (f+g)(x), and product as (fg)(x). In particular, f+g is the sum, fg is the product.
• Thus, to be clearer, it simply means

\begin{aligned} (f+g)(x) &=f(x)+g(x) \\(f g)(x) &=f(x) g(x)\end{aligned}

• When performing such functions, we have to be aware of the domain. For the sum and product to be defined, the domain of this combined function must be in both the domain of f and g.
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## 5.4 Composite Functions

#### Sketching Composite Functions

• Suppose we know f(x) and g(x), and we want to sketch h(x)=f(g(x)). Of course we can construct a table below (suppose g(x)=x^2+7,\ h(x)=x-1):

## 3.8 Transformation of Trigonometric Functions

#### Transformations with Sine and Cosine Graphs

• We can see that the graph of y=\cos x can simply be obtained by y=\sin x shifted \frac{\pi}{2} to the left. This is a translation, and the identity is not unfamiliar to us: \sin x=\cos \left(x-\frac{\pi}{2}\right).
• Now we shall look at graphs of simple transformations of these graphs. We shall start with y=a \sin (n t) and y=b \cos (n t). Naturally, n would be a dilation from the y-axis, and a (and b) is a dilation from the x-axis. The mechanism is similar, and is explained in A1 – Functions and Graphs/Relationship of Transformations . Here we will only provide a few simple examples.
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## 3.7 Transformations of Logarithmic Functions

#### 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:

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## 3.6 Transformations of Exponential Functions

#### 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:

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## 3.5 Transformation of Polynomial Functions

#### Transformations with Power Functions

• Recall that power functions has the form y=x^n, where is a real number. We have looked into cases where is a rational number, which includes positive and negative whole numbers and fractions.
• Transformation can be applied on power functions. The transformed equation simply take the general form of f(x)=a(b(x-c))^{n}+d, where a,\ b,\ c,\ d are real numbers.

i) If it is a reflection, we have b=-1 or a=-1.

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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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