1. 1 Functions الدكتور: كمال الهادي عبدالرحمن

2. 2 A3.1 A function is specified by: I.A Domain = allowed input values. II.A rule defined for each x in the domain to give a unique (output) value. For a function ƒ we use the notation y=ƒ(x) to give the value of the function. y is also called the image of x under ƒ. Examples : I.ƒ(x)=x 2 +2x, 0 ≤ x ≤ 2 II.g(x)=√x, x ≥ 0 III.h(t)=sint, 0 ≤ t ≤ 2л Domains are usually composed of intervals: Let a < b : The closed interval [a, b]={x :a ≤ x ≤ b}. The open intervals (a, b)={x :a < x < b}. Intervals can also be half open (half closed), or could be infinite. We write D(ƒ) or Dom(ƒ) for the domain of the function ƒ.

3. 3 Domain convention: When only the rule of the function is given the domain is taken to be the largest subset of R for which the rule is defined. Note in R division by zero and even roots of negative numbers are not defined. (Do activity 1.3). A3.2 Graphs of function: Graph of ƒ = {(x,ƒ(x): x є Dom(ƒ)} Examples : 1)Linear function :y=mx+b, graph is a straight line with slope m and y- intercept(0,b).See Figure 1 below: y x y=mx+c 0 ө ө Slope m= tan ө Fig 1

4. 4 2) Square function: y=ƒ(x)=x 2 Note that the figure is symmetric about y-axis. The graph is a parabola with vertex at 0. See Figure 2 below: 3) Reciprocal function :y=ƒ(x)=1/x, x ≠ 0. See Figure 3 below: Note as x ∞, we see that ƒ ( x )=1/x 0 So the line y=0 is an asymptote (horizontal asymptote). Note: symmetry about 0. Note also that the image set of a function can be read from its graph. We can also sketch functions with restricted domains. (Do activity 1.4). 0 1 2 3 4 12 1 -2 y=x 2 x y 1 -2 2 2 y x y=1/x Fig 3 Fig 2

5. 5 The absolute value or modulus function: It is the function defined by : ƒ(x)=|x|= With graph as in Figure 4 below. It has image set = [0,∞) (Do activity 1.5). A3.3 Quadratic functions: Given by y=ƒ(x)=ax 2 +bx+c,a≠0. Its graph could be obtained from the graph of x 2 (see fig 2) by certain geometric transformations summarized in the table below: x, x≥0 -x, x<0 Traslation or scaling of y=ƒ(x)Graph Horizontal translation by p units to the left if p>0 (to the right if p<0) y=ƒ(x+p) Vertical translation by q units upward if q>0(downwards of q<0) y=ƒ(x)+q y-Scaling with factor a. a>1gives expansion, 0<a<1gives contraction y=af(x) x-scaling with factor1/b, b>0.y=ƒ(bx) Gives reflection about x-axis.y=-ƒ(x) x y 0 y=|x| Fig 4

6. 6 Example: Graph of y= ƒ(x)=ax 2 +bx+c Solution : y=a[x 2 +(b/a)*x]+c =a[x+b/2a] 2 +c-b 2 /4a Star t from y=x 2,and then translate to the left by [b/2a] units, as shown in the figure. to get y= x+b/2a 2 Then do y-scaling by a units, see the figure below: to get y=a x+b/2a 2 yy x x -b/2ay=x 2 -b/2a y x

7. 7 Then vertical translation by c-b 2 /4a unit, as shown below: to get y=ax 2 +bx+c (Read frames(1-11)and do activity 2.5) Exercise:Draw the graph of y=2|x-1|-3, page 24. c-b 2 /4a -b/2/2a

8. 8 A3.4 Trigonometric functions We noted in A2 that sin and cos are periodic with period 2π that is: cos(t+2nπ)=cost,sin(t+2nπ)=sint, for all integers n. These with values in table A2.10 enable us to draw the graphs of sin and cos. See figure 5:x=cost, and figure 6:y=sint. π -π-π 2π2π3π3π-2π-3π x t 1 0 π -π-π 3π3π-2π-3π x t 1 0 2π2π Fig 5Fig 6

9. 9 We can also use left and right translations and x and y- scaling to draw graphs of: x=a cos(kt+c)+d,y=b sin(kt+c)+d. (See graphs in Book. A3 pp. 29-30) (Do activities 3.3 and 3.4). Graph of y=tan x could be obtained from the definition: y=sin x/cos x and some chosen values of tan x in(-π/2, π/2) and using periodicity of tan x (period=π) See Figure 7 below: -5/2π 5/2π-3/2π3/2π-1/2π1/2π y x 0 5 -5 y=tan x Note that graph has vertical asympototes at x=±(1/2)π,±(3/2)π,±(5/2)π,… Fig 7

10. 10 A3.5 Exponential functions : Let a>0 be a real number. If n is a positive integer then a n = a×a×…..×a, n times. For xє R,we define a x in steps: I.a 0 = 1. II.If n in a positive integer:a -n =1/a n. III.For p,q integers (q>0),we define a p/q =(√a) p..this defines rational exponents. IV.For xє R,a x is found to any desired accuracy by approximating x by a rational number p/q and a x ≈ a p/q Rules of exponents: I. a 0 =1 II.a -x =1/a x III.a x+y= a x.a y IV.a x-y =a x /a y V.(a x ) y =a xy n-times q

11. 11 Definition: For aє R,a > 0 and xє R ƒ(x)=a x is called the exponential function with base a. The graph of a x could be drawn from a table of values giving x and a x. Example: Graph of 2 x and 2 -x = (1/2) x. (Do activity3.5). Note: I.If a >1 then a x →∞ as n→∞. II.if 0<a<1 then a x →0 as n→∞. 21.510.50-0.5-1.5-2x 42.8321.4110.710.50.350.252x2x 0.350.50.7111.4122.834(1/2) x x y y=2 x y=(1/2) x 1 2 3 4 -212 0 Fig 8

12. 12 Figure 9 below shows the graph of a x in three cases: (a)a>1 (b) a=1 (c) 0<a<1 An important exponential function is that to the base e (defined in A2). e=lim﴾1+1/n﴿ n ≈ 2.718281…, an irrational number. The graph of e x lies between the graphs of 2 x and 3 x. It has a tangent line at (0,1) whose slope=1, as shown in Figure 10 below: 1 1 1 000 y yy xxx n→∞ 2 4 8 6 0,1)) 24-2 0 y x y=e x Slope of tangent line is 1 Fig 9 Fig 10

13. 13 A3.6 Inverse Functions: A function ƒ is said to be one to one(one-one,1-1) if: for all x 1,x 2 єDom(ƒ),if x 1 ≠x 2 thenƒ(x 1 )≠ƒ(x 2 ). Equivalently: if ƒ(x 1 )=ƒ(x 2 ) then x 1 =x 2. Examples of one – one functions are the monotone functions. These are: I.Increasing functions characterized by: for all x 1,x 2 єDom(ƒ):if x 1 <x 2 then ƒ(x 1 )<ƒ(x 2 ). II.Decreasing functions characterized by: for all x 1,x 2 єDom(ƒ):if x 1 ƒ(x 2 ). as illustrated by figure 11 below: a) Increasing b) decreasing c) not monotone Fig 11. (Do activity 4.1). x1x2x3 y x y x x1x2x1x2 y x

14. 14 Def: Let ƒ be1-1 function, we define its inverse functionƒ -1 by: x=ƒ -1 (y) y=ƒ(x). Note: Image set of ƒ=Dom(ƒ -1 ), image set of ƒ -1 =Dom(ƒ). Graph of y=ƒ -1 (x) and graph of y=ƒ(x) are symmetric about the line y = x, as shown in Fig 12 (a) Graph of y=ƒ(x) (b) graph of y=ƒ -1 (x) 45 0 y=ƒ(x) 45 0 line y=ƒ -1 (x) y x y x 11 1 Different scales Fig 12

15. 15 Example:y=g(x)=x 2 on[0,∞),g -1 (x)=√x has inverse function See Figure 13 below: y=g -1 (x)=√x y=g(x)y=x equal scales Fig 13

16. 16 To find the inverse function to y=ƒ(x): I.Write y=ƒ -1 (x) x=ƒ(y). II.In x=ƒ(y) solve for y in terms of x to get y=ƒ -1 (x).. Example: find ƒ -1 (x) if y=ƒ(x) =x 2 + 2, x≥0 and y ≥2. Solution: y=ƒ -1 (x) x=ƒ(y) = y 2 +2. y 2 =x-2 y=√(x-2). We take only the positive root since y ≥ 2. (Do Example 4.1 and activity 4.2).

17. 17 A3.7 Inverse trigonometric functions : a) ƒ(x)=sin x is 1-1 for xє[-π/2,π/2] with range=[-1,1]. Thus we can define its inverse function: sin -1 x(arcsin x) by: y=sin -1 x x=sin, y where xє[-1,1], yє[-π/2,π/2]. Graph of y=sin -1 x is obtained from that of y=sin x by reflection about y=x. See the two graphs below: Example: Since sin0=0,then sin -1 0=0. Since sinπ/2=1, then sin -1 1=π/2. y=ƒ -1 (x) =arcsin x y=ƒ (x)=sinx 1 1/2π -1/2π 1/2π-1/2π yy xx 1

18. 18 Also ƒ(x)= cos x is 1-1 in [0,π], with range[-1,1]. So we can define y=cos -1 x x=cos y, where xє[-1,1],yє[0,π]. Their graphs are below: Also y=tan -1 x x=tany,yє[-π/2,π/2]. Its graph is shown in the figure below: (Do activity 4.3). y=ƒ(x)=cosx 1/2π1 1 π x y 1/2π π y x 1 y=ƒ -1 (x)=arccos x y=ƒ(x)=tanx y=ƒ -1 (x)=arctan x y y xx 1/2π 1/4π -1/2π 1/4π11/2π-1/2π

19. 19 A3.8 Logarithms: If 1≠a>0, then function ƒ(x)=a x is 1-1 and has the graph below: (a) a> 1 (b) 0 <a <1 So we can define its inverse function ƒ -1 which is called logarithm to base a (denoted by log a, as follows: x=log a y y=a x,1≠a>0 yy 11 xx 00

20. 20 Example: since2 3 = 8, we have log 2 8=3., also since4=2 2,2=2 1, 1=2 0, and ½=2 -1, we have : l og 2 4=2, log 2 2=1, log 2 1=0, log 2 1/2=-1. Graph of y=log a x is obtained from graph of y=a x by reflection about y=x. See the graph below: 432 1 3 2 1 4 -2 y x y=x

21. 21 Properties of logarithms are obtained from properties of exponents as follows: a) log a 1=0, log a a=1 b) For x > 0, y> 0: i.log a (xy)=log a x+log a y ii.log a [x/y]= log a x-log a y c)For x >0, pєR, log a (x p )=p log a x d)log a x→∞ as x→∞ e)log a x→-∞ as x→0 Examples: Simplify: log a [ xy/z ] 2/3 =2/3log a [ xy/z ] =2/3[log a x+log a y-log a z] (Do activity 4.5).

22. 22 Special cases: 1)When, a = e≈ 2.718281….we get natural logarithms denoted by ln i.e. ln x = log e x. 2)When, a = 10, we get common logarithm denoted by log, i.e. logx = log 10 x. We can change from natural to common logarithms and conversely by:[ln y= log y.ln 10] This is obtained by writing x = log y y=10 x ֱ ln y=x ln 10 =log y. ln10 (Do activity 4.6 and exercises for section 4 in book A3).