03Kdomain

= Domain and Range =


 * Domain **


 * The ** domain ** of a function is the set of x-values for that function


 * For most functions, the domain is R, the set of all real numbers
 * this is written as
 * x Î R


 * For some functions we restrict the domain to some subset of R
 * this can be written as
 * 1 < x ≤ 10
 * or
 * x Î ( 1, 10 ]


 * Range **


 * The ** range ** of a function is the set of y-values for that function


 * For many functions, the range is R, the set of all real numbers
 * this is written as
 * y Î R


 * For the rest, the range will be some subset of R


 * for a parabola with the rule y = x 2 + 2,
 * the range would be
 * y ≥ 2
 * or
 * y Î [2, ¥ )
 * or
 * 2 ≤ y < ¥


 * Interval Notation **


 * The brackets used to describe a domain or range have a specific meaning


 * A square bracket [ ] means including the end value
 * x Î [3, 6] .. means .. 3 ≤ x ≤ 6


 * A smooth bracket means excluding the end value
 * x Î (3, 6) .. means .. 3 < x < 6


 * Infinity, ¥, should always be written with a smooth bracket


 * Graphing with a restricted Domain **


 * An open circle is used to show that the endpoint on a graph is __not__ included
 * A closed (or filled in) circle is used to show that the endpoint is included.


 * Example 1 **

... ... Sketch y = 2x – 1 in the domain x Î (1, 4]


 * Solution: **


 * See graph on the right
 * domain is x Î (1, 4] (or 1 < x ≤  4)
 * note the open circle at (1, 1)
 * note the closed circle at (4, 7)
 * range is y Î (1, 7]


 * Function Notation **


 * The formal definition of a function is written like this:

... ... f: (1, 4] → R, f(x) = 2x – 1


 * f is the name of the function
 * (1, 4] is the domain of the function
 * → R indicates that the function will take the domain and map it onto values from the set of Real numbers
 * f(x) indicates that the variable is x
 * 2x – 1 is the rule for the function.

Maximums and Minimums


 * A ** maximum point ** is the highest point on the graph (the upper end of the range)
 * A ** minimum point ** is the lowest point on the graph (the lower end of the range)


 * A ** Local Maximum ** is a turning point that is a maximum on a part of the graph but not necessarily the highest point on the entire graph
 * A ** Local Minimum ** is a turning point that is a minimum on a part of the graph but not necessarily the lowest point on the entire graph


 * Your calculator can locate local maximums and minimums on a graph
 * Sketch the graph in the Graphs&Tables part of the calculator
 * max and min commands are located in the ANALYSIS menu, GSOLVE submenu


 * Note **


 * Care should be taken when asked for the range of a cubic with a restricted domain
 * the upper and lower ends of the range could be at a local maximum or minimum
 * or they could be at the endpoints of the graph
 * A sketch is the best way to determine the correct answer.

Example 2

math \\ \text{Sketch } y = \frac{1}{3}x^3 + x^2 -3x-3 \text{ in the domain } x \in \big( -5\frac{1}{4}, \,2 \big] \\ \text{and state the range} math


 * Solution:**


 * ** With the aid of a calculator, draw the entire graph (in pencil), **
 * ** then mark in the endpoints and draw the desired section more clearly. **
 * ** __Erase__ the unwanted portions of the graph. **

X-intercepts

... ... x = –4.54, –0.83, 2.38

Y-intercepts

... ... y = –3

Turning Points

... ... Local maximum (–3, 6)

... ... Local minimum (1, –4.67)

Endpoints

... ... (–5.25, –8) ** {open circle} **

... ... (2, –2.33) ** {closed circle} **


 * Desired section is between the two endpoints (in dark blue)

Domain

... ... x Î (–5.25, 2]

Range


 * ** by observation of the graph **

... ... y Î (–8, 6]

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