Chapter two Functions

Chapter 2
Functions


2.1 Definition of the Function
A function from a set to a set is a rule that assigns a unique (single) element
ƒ( ) ∈ to each element ∈ , i.e.
: → ⟺∀ ∈ ∃ ∈ ℎ ℎ = ( ).
The set of all possible input values is called the domain of the function
= { ∈ ℝ: = ( )}.
The set of all values of ( ) as varies throughout is called the range of the
function
= ∈ : = ( ) ∀ ∈ .
The range may not include every element in the set .

Example 2.1 Find the domains and ranges of the following functions.
) = 1 ) =√
) =

) = √4 − ) = 1−
Solution

a) The formula = has been defined for all real number ∈ ℝ, so
= ℝ = (−∞, ∞).
The range of = is
= [0, ∞ )
because the square of any real number is nonnegative and every nonnegative

number y is the square of its own square root, = for ≥ 0.


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, b) The formula = is defined for every ∈ ℝ except = 0, we cannot divide

any number by zero.
= ℝ − {0} = (−∞, 0) ∪ (0, ∞).

The range of = is

= ℝ − {0} = (−∞, 0) ∪ (0, ∞).

c) The formula = √ gives a real −value only if ≥ 0.
= [0, ∞ )
The range of = √ is
= [0, ∞ ),


d) In = √4 − the quantity 4 − cannot be negative. That is, 4 − ≥ 0 or
≤ 4. The formula gives real −values for all ≤ 4.
= (−∞, 4]

The range of √4 − is
= [0, ∞ ),
the set of all nonnegative numbers.

e) The formula = √1 − gives a real y-value for 1 − ≥ 0 i.e.
= [−1,1],
since
1− ≥0⟺1≥ = | | ⟺ | | ≤ 1 ⟺ −1 ≤ ≤ 1.
The values of 1 − vary from 0 to 1 on the given domain, and the square
roots of these values do the same. The range of √1 − is
= [0,1].



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