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Functions and Their Graphs
DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ฦ(x) โ Y to each element x โ D. A symbolic way to say โ y is a function of x โ is by writing
y = ฦ(x) (โy equals ฦ of xโ)
The set D of all possible input values is called the domain of the function. The set of all values of ฦ( x ) as x varies throughout D is called the range of the function. The range may not include every element in the set Y.
EXAMPLE : Identifying Domain and Range Verify the domains and ranges of these functions.
Solution The formula y=x^2 gives a real y -value for any real number x , so the domain is (โโ,โ)The range of y=x^2 is [ 0 ,โ) because the square of any real number is nonnegative and every nonnegative number y is the square
of its own square root, y=( ๐ฆ )^2 for yโฅ 0.
The formula y=1/x gives a real y -value for every x except x=0 We cannot divide any number by zero. The range of y=1/x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y=1/(1/y).
The formula y = ๐ฅ gives a real y -value only if x โฅ 0 The range of
y = ๐ฅ is [ 0 ,โ) because every nonnegative number is some numberโs square root (namely, it is the square root of its own square).
In y = 4 โ ๐ฅ the quantity 4 โ ๐ฅ cannot be negative. That is, 4 โ ๐ฅ โฅ 0 or x โค 4 The formula gives real y -values for all x โค 4 The range of
4 โ ๐ฅ is [ 0 ,โ) the set of all nonnegative numbers.
The formula y = 1 โ ๐ฅ^2 gives a real y -value for every x in the closed interval from โ 1 to 1. Outside this domain, 1 โ ๐ฅ^2 is negative and its square root is not a real number. The values of 1 โ ๐ฅ^2 vary from 0 to 1 on the given domain, and
the square roots of these values do the same. The range of 1 โ ๐ฅ^2 is [0, 1].
Piecewise-Defined Functions
Sometimes a function is described by using different formulas on different parts of its domain. One example is the absolute value function
EXAMPLE : Graphing Piecewise-Defined Functions The function
EXAMPLE : The Least Integer Function The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted ๐ฅ.
The graph of the least integer function ๐ฆ = ๐ฅ lies on or above the line ๐ฆ = ๐ฅ so it provides an integer ceiling for x
EXAMPLE : graph the function y =x 2 ๐ฅ โ 1 [-3,2)
Solution
The following table summarizes the formulas and domains for the various algebraic combinations of the two functions. We also write ฦ. g for the product function ฦ g.
H.W
Functions find the domain and range of each function.
Piecewise-Defined Functions Graph the functions
