1. CALCULUS
AND
ANALYTIC GEOMETRY
CS 001
LECTURE 02

2. Function

3. Function

4. Definition of Function
The element x of D is the argument of f. The set D is the domain of the function. The element y of E is the value of f at x (or the image of x under f ) and is denoted by f (x), read “f of x.”
The range of f is the subset R of E consisting of all possible values f (x) for x in D.
Note that there may be elements in the set E that are not in the range R of f.

5. Definition of Function
Consider the diagram in Figure 2.
The curved arrows indicate that the elements f (w), f (z), f (x) and f (a) of E correspond to the elements w, z, x, and a of D.
Figure 2

6. Definition of Function
To each element in D there is assigned exactly one function value in E; however, different elements of D, such as w and z in Figure 2, may have the same value in E.
The symbols
f : D  E, and
signify that f is a function from D to E, and we say that f maps D into E. Initially, the notations f and f (x) may be confusing. Remember that f is used to represent the function.

7. Definition of Function
It is neither in D nor in E. However f (x), is an element of the range R—the element that the function f assigns to the element x, which is in the domain D.
Two functions f and g from D to E are equal, and we write
f = g provided f (x) = g(x) for every x in D.
For example, if g(x) = (2x2 – 6) + 3 and f (x) = x2 for every x in , then g = f.

8. Example 1 – Finding function values
Let f be the function with domain such that f (x) = x2 for every x in .
(a) Find f (– 6), f ( ), f (a + b), and f (a) + f (b), where a and b are real numbers.
(b) What is the range of f ?
Solution:
(a) We find values of f by substituting for x in the equation f (x) = x2:
f (– 6) = (– 6)2 = 36
f ( ) = ( )2 = 3

9. Example 1 – Solution f (a + b) = (a + b)2 = a2 + 2ab + b2
cont’d
f (a + b) = (a + b)2 = a2 + 2ab + b2
f (a) + f (b) = a2 + b2
(b) By definition, the range of f consists of all numbers of the form f (x) = x2 for x in .
Since the square of every real number is nonnegative, the range is contained in the set of all nonnegative
real numbers.
Moreover, every nonnegative real number c is a value of f, since f ( ) = ( )2 = c. Hence, the range of f is
the set of all nonnegative real numbers.

10. Example 2 – Finding function values
Let g (x) =
(a) Find the domain of g.
(b) Find g(5), g(–2), g(–a), and –g(a).
Solution:
(a) The expression is a real number if and only if the radicand 4 + x is nonnegative and the denominator 1 – x is not equal to 0.
Thus, exists if and only if
4 + x  0 and 1 – x ≠ 0

11. Example 2 – Solution x  – 4 and x ≠ 1.
or, equivalently,
x  – and x ≠ 1.
We may express the domain in terms of intervals as [– 4, 1)  (1, ).
(b) To find values of g, we substitute for x:
cont’d

12. Example 2 – Solution
cont’d

13. Definition of Function
Graphs are often used to describe the variation of physical quantities. For example, a scientist may use the graph in Figure 5 to indicate the temperature T of a certain solution at various times t during an experiment.
Figure 5

14. Definition of Function
The sketch shows that the temperature increased gradually from time t = 0 to time t = 5, did not change between t = 5 and t = 8, and then decreased rapidly from t = 8 to t = 9.
Similarly, if f is a function, we may use a graph to indicate the change in f (x) as x varies through the domain of f.
Specifically, we have the following definition.

15. Definition of Function
In general, we may use the following graphical test to determine whether a graph is the graph of a function.
The x-intercepts of the graph of a function f are the solutions of the equation f (x) = 0. These numbers are called the zeros of the function.
The y-intercept of the graph is f (0), if it exists.

16. Example 3 – Sketching the graph of a function
Let f (x) =
(a) Sketch the graph of f.
(b) Find the domain and range of f.
Solution:
(a) By definition, the graph of f is the graph of the equation
y =
The following table lists coordinates of several points on the graph.

17. Example 3 – Solution
Plotting points, we obtain the sketch shown in Figure 7. Note that the x-intercept is 1 and there is no y-intercept.
Figure 7

18. Example 3 – Solution
(b) Referring to Figure 7, note that the domain of f consists of all real numbers x such that x  1 or, equivalently, the interval [1, ).
The range of f is the set of all real numbers y such that y  0 or, equivalently, [0, ).

19. Definition of Function
In general, we shall consider functions that increase or decrease on an interval I, as described in the following chart, where x1 and x2 denote numbers in I.
Increasing, Decreasing, and Constant Functions

20. Definition of Function

21. Definition of Function
An example of an increasing function is the identity function, whose equation is f (x) = x and whose graph is the line through the origin with slope 1.
An example of a decreasing function is f (x) = – x, an equation of the line through the origin with slope – 1. If f (x) = c for every real number x, then f is called a constant function.
We shall use the phrases f is increasing and f (x) is increasing interchangeably. We shall do the same with the terms decreasing and constant.

22. Example 4 – Using a graph to find domain, range, and where a function increases or decreases
Let f (x) =
(a) Sketch the graph of f.
(b) Find the domain and range of f.
(c) Find the intervals on which f is increasing or is decreasing.
Solution:
(a) By definition, the graph of f is the graph of the equation
y =
We know from our work with circles that the graph of x2 + y2 = 9 is a circle of radius 3 with center at the origin.

23. Example 4 – Solution
Solving the equation x2 + y2 = 9 for y gives us y = 
It follows that the graph of f is the upper half of the circle, as illustrated in Figure 8.
Figure 8

24. Example 4 – Solution
(b) Referring to Figure 8, we see that the domain of f is the closed interval [–3, 3], and the range of f is the interval [0, 3].
(c) The graph rises as x increases from –3 to 0, so f is increasing on the closed interval [–3, 0].
Thus, as shown in the preceding chart, if x1 < x2 in [–3, 0], then f (x1) < f (x2) (note that possibly x1 = –3 or x2 = 0)

25. Example 4 – Solution
cont’d
The graph falls as x increases from 0 to 3, so f is decreasing on the closed interval [0, 3].
In this case, the chart indicates that if x1 < x2 in [0, 3], then f (x1) > f (x2) (note that possibly x1 = 0 or x2 = 3)

26. Definition of Function
The following type of function is one of the most basic in algebra.
The graph of f in the preceding definition is the graph of y = ax + b, which, by the slope-intercept form, is a line with slope a and y-intercept b.
Thus, the graph of a linear function is a line.

27. Example 6 – Sketching the graph of a linear function
Let f (x) = 2x + 3.
(a) Sketch the graph of f.
(b) Find the domain and range of f.
(c) Determine where f is increasing or is decreasing.
Solution:
(a) Since f (x) has the form ax + b, with a = 2 and b = 3, f is a linear function.

28. Example 6 – Solution
The graph of y = 2x + 3 is the line with slope 2 and y-intercept 3, illustrated in Figure 10.
Figure 10

29. Example 6 – Solution
(b) We see from the graph that x and y may be any real numbers, so both the domain and the range of f are
(c) Since the slope a is positive, the graph of f rises as x increases; that is, f (x1) < f (x2) whenever x1 < x2. Thus, f is increasing throughout its domain.

30. Some Standard Real Functions (Constant Function)Y X
(0, c)
f(x) = c
Domain = R
Range = {c}

31. Identity Function X Y O
450
I(x) = x
Domain = R
Range = R

32. Modulus Function
f(x) = x
f(x) = - x O X Y