1. Function Graph

Home , Even and odd functions

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1. FUNCTION GRAPH

1.1 Functions; Domain and Range

The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The area of a circle depends on the radius of the circle. The distance an object travels at constant speed along a straight-line path depends on the elapsed time. In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x. We say that “y is a function of x” and write this symbolically as

y = ƒ(x) (“y equals ƒ of x”).

- ƒ represents the function - x is the independent variable representing the input value of ƒ - y is the dependent variable or output value of ƒ at x.

- The set D of all possible input values (X) is called the domain of the function. - The range may not include every element in the set (Y). - A value of ƒ(x) as x varies throughout D is called the range of the function. - mostly , Range and domain are sets of real numbers interpreted as points of a coordinate line.

A function 푓 is like a machine that produces an output value 푓(푥) in its range whenever we feed it an input value 푥 from its domain (Figure 1.1 . The function keys

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019) on a calculator give an example of a function as a machine.

Example :- Let’s verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense.

No Function Domain (x) Range (y)

1 푦 = 푥2 (−∞, ∞) [0, 푞) 2 푦 = 1/푥 (−∞, 0) ∪ (0, ∞) (−∞, 0) ∪ (0, ∞)

3 푦 = √푥 [0, ∞) [0, ∞) 4 푦 = √4 − 푥 (−∞, 4] [0, ∞)

5 푦 = √1 − 푥2 [−1, 1] [0, 1]

Solution:- 1- The formula 푦 = 푥2 gives a real 푦‐value for any real number 푥. The range of 푦 = 푥2 is [0, ∞) because the square of any real number is nonnegative . 2- The formula 푦 = 1/푥 gives a real 푦‐value for every 푥 except 푥 = 0 because if x =0 result will be ∞. Thus, range of 푦 = 1/푥, the set of reciprocals of all nonzero real numbers. 3- The formula 푦 = √푥 gives a real 푦‐value only if 푥 ≥ 0. The range of 푦 = √푥 is

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

[0, ∞) .

4- In 푦 = √4 − 푥, the quantity 4 −푥 cannot be negative. That is, 4 −푥 ≥ 0, or 푥 ≤

4. The formula gives real 푦‐values for all 푥 ≤ 4. The range of √4 − 푥 is [0, ∞), the set of all nonnegative numbers.

5- The formula 푦 = √1 − 푥2 gives a real 푦‐value for every 푥 in the closed interval from −1 to 1. Outside this domain, 1 −푥2 is negative and its square root is not a real number. The values of 1 −푥2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of

√1 − 푥2 is [0, 1].

1.2 Represented Function By Graphs

If 푓 is a function with domain 퐷, its graph consists of the points in the Cartesian plane whose coordinates are the input‐output pairs for 푓. In set notation, the graph is {(푥, 푓(푥))|푥 ∈ 퐷}

Let see how will be graphs of the function

푓(푥) = 푥 + 2

Where the 푦 = 푓(푥) the represented by Y-axis , and the x will represented the input as X-axis in the coordinates of the (푥, 푦) , the result show in fig. 1.3

x 풚 = 풙 + ퟐ 2 4 0 2 -2 0

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

the previous example conclude the height in positive or negative value of the carve or line is F(x), while the x-axis change as time response or input value of the main function, as shown the figure 1.4.

Figure 1.4

Example:- Graph the function 푦 = 푥2 over the interval [−2, 2].

Solution:- Make a table of xy‐pairs that satisfy the equation = 푥2. Plot the points (푥, 푦) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5).

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

How do we know that the graph of 푦 = 푥2 doesn’t look like one of these curves?

1.3 Representing a Function Numerically We have seen how a function may be represented algebraically by a formula (the area function) and visually by a graph (Example 2 . Another way to represent a function is numerically, through a table of values. Numerical representations are often used by engineers and scientists. The data in Table 1.1 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The data points (푡, 푝) from the table, we obtain the graph shown in Figure 1.6.

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.4 Functions Specified by Equations

Not every curve in the coordinate plane can be the graph of a function. A function 푓 can have only one value 푓(푥) for each 푥 in its domain, so no vertical line can intersect the graph of a function more than once. If 푎 is in the domain of the function 푓, then the vertical line 푥 = 푎 will intersect the graph of 푓 at the single point (푎, 푓(푎)) . Such shown in these examples below:-

Functions Specified by Equations - If in an equation in two variables, we get exactly one output (value for the dependent variable) for each input (value for the independent variable), then the equation specifies a function. The graph of such a function is just the graph of the specifying equation. 6

- If we get more than one output for a given input, the equation does not specify a function.

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example :- Determine which of the following equations specify functions with independent variable x. (A) 4y - 3x = 8, x a real number (B) y2 - x2 = 9, x a real number (A) Solving for the dependent variable y, we have 4y - 3x = 8 4y = 8 + 3x y = 2 + (3/4)x (1) thus from the result show the equation specifies a function.

(B) Solving for the dependent variable y, we have y2 - x2 = 9 y2 = 9 + x2

y =± √9 + 푥2 ( 2)

Since is always a positive real number for any real number x and since each positive real number has two square roots. As example if x = 4 then equation (2) is satisfied for y = 5 and y = -5. Thus, equation (2) does not specify a function. Note*: each positive real number √ N has two square roots: the principal square root, and -√N, the negative of the principal square root

## vertical line for test function The result of the previous example can show by using the graph the equation where, if the test line intersect the graph in one pine mean that equations specify functions, while, if line intersect the graph by tow point that mean the equation does not specify a function. As shown in the figure below

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Figure 1.7

1.5 Piecewise‐Defined Functions Sometimes a function is described by using different formulas on different parts of its 푥 domain. Such as the example shown below which is plotted in figure 1.7:- 푥, 0 < 푥 The absolute value |푥| = {푥, 푥 < 0 푥 푥 = 0

Figure 1.7

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example:- sketch the function shown below :

−푥, 푥 < 0 푓(푥) = {푥2, 0 ≤ 푥 ≤ 1 1, 푥 > 1

The result of this function is shown in figure 1.8

Figure 1.8

1.5.1 Greatest Integer Function Firstly let define the Greatest Integer Function. The Greatest Integer Function the integer floor function is denoted by y = ⌊푥⌋. For all real numbers of x, the greatest integer function returns the largest integer or less than x. While, smallest integer is greater than or equal to 푥 is called the least integer function or the integer ceiling. It is denoted⌈푥⌉.

Example :- sketch the function shown below that related to greatest integer function 푓(푥) = ⌊푥⌋ , Solation :- the table and figure 1.9 shows result of the this function.

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Figure 1.9

1 Example :- Sketch a graph of y=⌊ 푥⌋ . 2 Solution :- the table and the figure 1.10 show the solution of this function.

x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ½ x -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1 ⌊ 푥⌋ -1 -1 -1 -1 0 0 0 0 1 2

Figure 1.10

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example :- sketch the function 푓(푥) = ⌈푥⌉ that related to least Integer function. Sol.:- the table and figure 1.11 showed the solution of this example.

x 2.4 2 1.9 0.2 0 -0.3 -1.2 -1.7 -2.2 ⌈푥⌉ 3 2 2 1 0 0 -1 -1 -2

Figure 1.11

1.6 Increasing and Decreasing Functions If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing. If the graph descends or falls as you move from left to right, the function is decreasing.

Let 푓 be a function defined on an interval 퐼 and let 푥 and 푥 be any two points 1 2 in 퐼.

1. If 푓(푥 ) > 푓(푥 ) whenever 푥 < 푥 , then 푓 is said to be increasing on 퐼. 2 1 1 2 2. If 푓(푥 ) < 푓(푥 ) whenever 푥 < 푥 , then 푓 is said to be decreasing on 퐼. 2 1 1 2

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example:- The function graphed in Figure 1.8 is decreasing on (−푞, 0] and in‐ creasing on [0, 1]. The function is neither increasing nor decreasing on the interval [1 , q) because ofthe strict inequalities used to compare the function values in the definitions.

1.7 Even Functions and Odd Functions: Symmetry The graphs of even and odd functions have characteristic symmetry properties.

DEFINITIONS - A function 푦 = 푓(푥) is an even function of x if 푓(−푥) = 푓(푥) - odd function of x if 푓(−푥) = −푓(푥) ,for every 푥 in the function’s domain.

 The names even and odd come from powers of 푥. If 푦 is an even power of 푥, as in 푦 = 푥2 or 푦 = 푥4, it is an even function of 푥 because (−푥)2 = 푥2 and (−푥)4 = 푥4. This type of the function are symmetric y-axis, where, the point (푥, 푦) will be in point (−푥, 푦). As shown in figure 1.12a  If 푦 is an odd power of 푥, as in 푦 = 푥 or 푦 = 푥3, it is an odd function of 푥 because (−푥)1 = −푥 and (−푥)3 = −푥3. This type of the function are symmetric about the origin point , where, the point (푥, 푦) will be in point (−푥, −푦). as shown in figure 1.12b

Figure 1.12a

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Figure 1.12b

Example 8: evaluated the function to describe and sketch to show (even or odd) type 1- 푓(푥) = 푥2 2- 푓(푥) = 푥2 + 1 3- 푓(푥) = 푥 4- 푓(푥) = 푥 + 1 Solutions:- 1- 푓(푥) = 푥2 Even function: (−푥)2 = 푥2 for all 푥; symmetry about 푦‐axis. The graph of 푦 = 푥2 shown in figure 1.13a 2- 푓(푥) = 푥2 + 1 Even function: (−푥)2 + 1 = 푥2 + 1 for all 푥; symmetry about 푦‐axis (Figure 1. 13푎). 3- 푓(푥) = 푥 Odd function:(−푥) = −푥 for all 푥; symmetry about the origin. The graph shown in figure 1.13b. 4- 푓(푥) = 푥 + 1 Not odd: 푓(−푥) = −푥 + 1, 푏푢푡 − 푓(푥) = −푥 − 1. The two are not equal. Because the adding of 1 to function thus, the symmetry about the origin is lost Not even: (−푥) + 1 ≠ 푥 + 1 for all 푥 ≠ 0 (Figure 1. 13푏 .

Figure 1.13

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8 Common Functions

A variety of important types of functions are frequently encountered in calculus. We identify and briefly describe them here.

1.8.1 Linear function The general formal of the liner function is

푓(푥) = 푚푥 + 푏 where 푚 and 푏 are constant values - If b= 0 function will pass through (0,0) as shown in Figure 1.14a - Identity function is 푓(푥) = 푥 , where 푚 = 1 and 푏 = 0. - Constant functions result when the slope 푚 = 0 (Figure 1. 14푏)

Figure 1.14 (a) linear function with with slope 푚 (b) A constant function with slope 푚 = 0.

DEFINITION: - Two variables 푦 and 푥 are proportional (to one another) if one is always a constant multiple of the other; that is, if 푦 = 푘푥 for some nonzero constant 푘.

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8.2 POWER FUNCTION The general formal of the power function is

푓(푥) = 푥푎, where 푎 is a constant, the value of 푎 should as : (a) 풂 = 풏, a positive integer. 푓(푥) = 푥푛, for 푛 = 1, 2, 3, 4, 5 ….. Function properties - Line pass through (1, 1) and (0, 0). - Tend to flatten in 푥‐axis on (−1, 1) , and rise more steeply for |푥| > 1. - The 푎 –even is decreasing function during (−∞, 0] ,and increasing on[0, ∞); - The 푎 –odd is increasing function real line (−∞, ∞), as shown in figure 1.15.

Figure 1 .15

(b) 풂 = −ퟏ or 풂 = −ퟐ. - functions are defined for all 푥 ≠ 0 - The graph of 푦 = 1/푥 is the hyperbola = 1, - The graph of 푦 = 1/푥2 also approaches the coordinate axes. As shown in Figure 1.16.

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Figure 1 .16

ퟏ ퟏ ퟑ ퟐ (c) 풂 = , , , and ퟐ ퟑ ퟐ ퟑ The functions 푓(푥) = 푥1/2 = 2√푥 , 푔(푥) = 푥1/3 = 3√푥 and 푦 = 푥3/2 = (푥1/2)3 are the square root and cube - The domain of the square root function is [0, ∞). - The cube root function is defined for all real 푥. As shown in Figure 1.17

Figure 1 .17

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8.3. Polynomial Function A function 푝 is a polynomial if

풏 풏−ퟏ 풑(풙) = 풂풏풙 + 풂풏−ퟏ풙 + …… + 풂ퟏ풙 + 풂ퟎ

- Polynomials of degree 2 (quadratic functions) written as

p(x) = ax2 + bx + c

- Polynomials of degree 3 (cubic functions) written as

p(x) = ax3 + bx2+ cx+ d,

Ex: x3 + 5x2 + 3 x5 - x3 + x0.5 ………….. etc. the graph of polynomial shown in figure 1.18

Figure 1 .18

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example: sketch the function 푓(푥) = (푥 − 2)(푥 + 1) Solution: shown in figure 1.19

2 푓(푥) = 푦 = 푥 + 푥 − 2푥 − 2 푦 = 푥2 − 푥 − 2 compare with 푦 = 푎푥2 + 푏푥 + 푐

The vertex of the a quadratic equation = −푏/2푎 The vertex = 푥 = −푏/2푎 , 푥 = −(−1)/2∗1 = 1/2

y= (1/2)2 -1/2 -2 =-9/4 vertex (ퟏ/ퟐ, −ퟗ/ퟒ) points of intercept at 푥 = 0 푦 = −2 at y = 0 0 = (푥 − 2)(푥 + 1) 푥 = 2 (2,0) 푥 = −1 (−ퟏ, ퟎ)

Figure 1 .19

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8.4 Trigonometric Functions

The six basic trigonometric functions are

Sine sin x = a/c Cosine cos x = b/c Tangent tan x = a/b = sin x/cos x

Cotangent cot x = b/a = cos x/sin x Secant sec x = c/b = 1/cos x Cosecant csc x = c/a = 1/sin x

 Identifies

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

- The graphs of the sine, cosine and tangent functions are shown in Figure 1.20.

Figure 1 .20

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

 Hyperbolic Functions

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8.5 Exponential Functions Exponential functions formula is 푓(푥) = 푎푥, where 푎 > 0 and 푎 ≠ 1, 푎 = constant value - Domain (−∞, ∞) and range (0, ∞) . - In fact, natural exponential function (ex ) , where e is constant number, e =2.71828 - An exponential function never assumes the value 0 , as shown in Figure 1.21.

Figure 1 .21

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

Example : We illustrate the flowing functions using the rules for exponents

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

1.8.6 Logarithmic Functions The functions describe as

푓(푥) = log푎푥 where 푎 ≠ 1 ,is a positive constant

- It is the inverse functions of the exponential functions - In each case the domain is (0, ∞) and the range is (−∞, ∞) . as shown in Figure 1.19.

Figure 1 .19

 NATURAL LOGARITHMS The logarithm with base is called the natural logarithm and has a special notation:

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

푥 푦 = 푙푛푥 0 ∞ 1 0 2 0.693 3 1.098 4 1.386 5 1.609 −1 ∞ 0.9 −0.105 0.5 −0.693 0.2 −1.609 0. 1 −2.302

Example: Classify the following functions as one of the types of functions that we have discussed.

1.8.7 Algebra of functions Let f is a function of x then we get f(x) and g is a function of x also we get g(x)

Df is the domain of f(x) Dg is the domain of g(x) Then: 25

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

f+g = f(x) + g(x) and Df ∩ Dg f – g = f(x) - g(x) f.g = f(x) . g(x) and the domain is as same before if f/g then Df ∩ Dg but g(x) ≠ 0 if g/f then Dg ∩ Df but f(x) ≠ 0

and Dfog = {x: x ∈ Dg, g(x) ∈ Df}

where fog(x) = f(g(x)) also called the composition of f and g

Example: Find fog and gof if 풇(x)= √1 − 푥 and g(x)= √5 + 푥 Solution:-

(fog)x= f(g(x)) = f(√5 + 푥 ) = √1 − √5 + 푥 (1-x) ≥ 0 then x ≤ 1 Df: x≤1 5+x ≥ 0 then x ≥ -5 Dg: x≥-5

D fog = {x: x ≥ -5, √5 + 푥 ≤1} = {x: -5 ≤ x ≤-4}

Example: If 푓(푥) = √푥 and 푔(푥) = √1 − 푥 Find:

푓 + 푔, f‐g, g‐f, 푓표푔, 푓/푔, 푔/푓 then graph 푓표푔 and also 푓 + 푔. Solution

푓(푥) = √푥 domain 푥 ≥ 0 푔(푥) = √1 − 푥 domain 푥 ≤ 1

푓 + 푔 = (푓 + 푔)푥 = √푥 + √1 − 푥 domain 0 ≤ 푥 ≤ 1 or [0,1] f‐g = √푥 − √1 − 푥 domain 0 ≤ 푥 ≤ 1 g‐f = √1 − 푥 − √푥 domain 0 ≤ 푥 ≤ 1

Al Muthanna University Math I College of Engineering Dr. Moneer Ali Lilo Department(s): chemical Eng. Semester I (2018-2019)

푓표푔 = 푓(푥)푔(푥) = 푓(푔(푥)) = 푓(√1 − 푥) = √√1 − 푥 = 4√1 − 푥 domain (−∞, 1] (why?)

푥 푓/푔 = 푓(푥)/푔(푥) = √ domain (−∞, 1] 1−푥

1−푥 푔/푓 = 푔(푥)/푓(푥) = √ domain (0,1] 푥

This chapter based on many reference [1][2][3] reference : [1] G. B. Thomas, calculus, Twelfth Ed. . [2] “Chapter 1, calculus,” Tikrit Univ. Coll. Eng. Civ. Eng. Dep., pp. 1–54, 2008. [3] “University of Anbar College of Engineering Department(s): Dams & Water Resources Eng. Electrical Eng .,” pp. 1–113, 2018.