Type I. There is no restriction on the inputs.

For example, f(x)=ax²+bx+c (quadratic function or any polynomial functions), (absolute function), f(x)=mx+b. The domain of the function is all real numbers.

Type II. Fractional functions

Example: If the domain of f = all reals the reason is that the input (x) can't be 1 or

Exercises. Find the domain of the following functions:

1. [all reals except x=-1, 3]
2. [all reals except -1].

Type III. Radical functions

Example: If find the domain of f. The key is that the quantity inside the even radical should be positive, i.e. which amounts to solve an inequality. (so you need to review techniques of solving inequalities.) Dom (f)

Example: If then find the domain of f. There are two restrictions on f, one is from the sqare root, and Thus the domain of f

Exercises: Find the domain of the following functions:

Horizontal Shifting:

Note that y=f(x+a) is a horizontal shifting of y=f(x).

Example: Note that is a horizontal shifting of f(x)=x², is shifted to the right one unit from y=x².

Vertical Shifting:

Note that y=f(x)+a is a vertical shifting of y=f(x). If a>0, then it is shifted up, and if a<0, the graph is shifted down.

Example: Note that is shifted down 2 units from

Exercises: Use the shifting techniques to sketch the graph of the following functions:

 
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