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1. Definition of function
Function is a relation whereby every element in the domain has one unique image in the range. There are four types of relations:
- one-to-one
- one-to-many
- many-to-one
- many-to-many
Based on the definition, one-to-one relation and many-to-one relation are functions. Many-to-one relation and many-to-many relation are not functions.
2. Notation of function
The common way of writing a function is
where
x is the independent variable, y is the dependent variable and f is the function name.
The independent variable x is sometimes called the input and the dependent variable y is sometimes called the output.
It is also common to write the function without a function name. For instance, you can write as y=fx=x or as y=x.
Example
y=f(x)=x is a one-to-one function because every element of x has a unique value of y. For example, when x=-2, y=-2, when x=3, y=3, when \(x={1\over2}\) and \(y={1\over2}\) and so on.
\(y=f(x)=x^2 \) is a many-to-one function because two different values of x can be mapped onto the same y. For example, when x= -2, \(y=(-2)^2\) , y=-2=4 and when x=2, \(y=(2)^2=4 \) and that is f-2=f2=4.
3. Domain and range
Domain of a function is the set of all the values of the independent variable,
x for which the function is defined. Range of a function is the set of all the values of the dependent variable, y that corresponds to the domain. You can use capital D to denote domain and capital R to denote range of a function.
Example
1. The domain for y=f(x)=x is all real numbers of x, written as D: x∈R. Since y=x, the range is also all real numbers, written as R: y∈R.
2. The domain for y=f(x)=x2 is all real numbers because any real number can be squared. The range consists of y=0 and all positive values of y because the square of any number cannot be negative. Note that the square of 0 is still 0. Thus, domain D:x∈R and R: y≥0.
4. Inverse functions
Inverse of function
f(x), written as \(f^{-1}(x)\), is obtained by interchanging x and y.
Example
1. Find the inverse function for the following functions:y=f(x)=x+2
Interchanging x and y in function y=x+2, you get x=y+2. Then, y=x-2.
∴ \(f-^{1}(x)=x-2\)
2. y=f(x)=2x-5
Interchanging x and y in function y=2x-5, you get x=2y-5.
Then, 2y=x+5 and \(y={1\over2}(x+5)\)
∴ \(f^{-1}x={1\over2}(x+5) \)
3. \(y=f(x)={x\over{x-2}}\)
Interchanging x and y in function \(y={x\over{x-2}}\) you get \(x={y\over{y-2}}\).
Then,
xy-2x=y
xy-y=2x
y(x-1)=2x
\(y={2x\over{x-1}}\)
∴ \(f^{-1}(x)={2x\over{(x-1)}}\)
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