Functions Review

1. Functions Review

2. What is a function • A function is a set of numbers in which every x has a y value • The x values are called the functions DOMAIN • The y functions are called the called the RANGE • Functions are usually written as f(x) (but they can also be written as g(x), h(x) etc) • If you have to evaluate a function, remember it’s a substitution

3. Evaluating functions

4. Domain • Hey, Remember that we said that Domain was the x values of an equation so • Look for which values you can graph, • Look for x values which make the bottom of a fraction zero – Not allowed • Look for values which give you the square root of a negative number – Not allowed either

5. Find the domain of the following functions: A) B) Domain is all real numbers but

6. C) Square root is real only for nonnegative numbers.

7. Graph of a function The graph of the function f(x)is the set of points (x,y) in the xy-plane that satisfy the relation y = f(x).

8. Domain and Range from the Graph of a function Domain = {x / or } Range = {y / or }

9. Determine the domain, range, and intercepts of the following graph. y 4 (2, 3) (10, 0) 0 (4, 0) (1, 0) x (0, -3) -4

10. Theorem Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

11. y x Not a function.

12. y x Function.

13. Is this a graph of a function? y 4 (2, 3) (10, 0) 0 (4, 0) (1, 0) x (0, -3) -4

14. Even functions A function f is an even function if for all values of x in the domain of f. Example: is even because

15. Odd functions A function f is an odd function if for all values of x in the domain of f. Example: is odd because

16. Determine if the given functions are even or odd

17. Graphs of Even and Odd functions The graph of an even function is symmetric with respect to the x-axis. (i.e. you can flip it with an axis of symmetry SIDEWAYS) The graph of an odd function is symmetric with respect to the origin. (i.e. you can rotate it 180 and get the same graph!)

19. Determine if the function is even or odd?

20. Determine if the function is even or odd?

21. Determine if the function is even or odd?

22. Continuous Functions • Consider the graph of f(x) = x3 • We can see that there are no "gaps" in the curve. Any value of x will give us a corresponding value of y. We could continue the graph in the negative and positive directions, and we would never need to take the pencil off the paper. • Such functions are called continuous functions

23. Discontinuous Functions • Consider the graph of y= 2/x: • We note that the curve is not continuous at x = 1. Such functions are called discontinuous functions • In simple English: The graph of a continuous function can be drawn without lifting the pencil from the paper.

24. Example of limits at infinity • The function can converge The function converges to a single value (1), called the limit of f. We write limx + f(x) = 1

25. Example of limits at infinity • The function can converge The function converges to a single value (0), called the limit of f. We write limx + f(x) = 0

26. Example of limits at infinity • The function can diverge The function doesn’t converge to a single value but keeps growing. It diverges. We can write limx + f(x) = +

27. Example of limits at infinity • The function can diverge The function doesn’t converge to a single value but its amplitude keeps growing. It diverges.

28. Example of limits at infinity • The function may neither converge nor diverge!

29. Example of limits at infinity • The function can do all this either at + infinity or - infinity The function converges at - and diverges at + . We can write limx + f(x) = + limx - f(x) = 0

30. Example of limits at infinity • The function can do all this either at + infinity or - infinity The function converges at + and diverges at -. We can write limx + f(x) = 0

31. Examples of limits at x=0 (x becomes very small!) • The function can have asymptotes (it diverges). The limit at 0 doesn’t exist…

32. Examples of limits at x=0 • The function can have a gap! The limit at 0 doesn’t exist…

33. Examples of limits at x=0 • The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

34. Examples of limits at x=0 • But most functions at most points behave in a simple (boring) way. The function has a limit when x tends to 0 and that limit is 0. We write limx  0 f(x) = 0

35. Evaluating Limits • Have a look at this video here. It’s not bad. • (the last bit on division might mean something to 3unit students but not 2 unit!)

36. B c a Side Opposite a Sinθ= Cos θ= Tan θ= = Hypothenuse c Side Adjacent b ө = A c Hypothenuse C b Side Opposite a = Side Adjacent b A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent. Let us look at this triangle… Given the assigned letters to the sides and angles, we can determine the following trigonometric functions. The Cosecant is the inversion of the sine, the secant is the inversion of the cosine, the cotangent is the inversion of the tangent. With this, we can find the sine of the value of angle A by dividing side a by side c. In order to find the angle itself, we must take the sine of the angle and invert it (in other words, find the cosecant of the sine of the angle).

37. B 10 α 6 β θ A C 8 B α 2 β 34º A C Try finding the angles of the following triangle from the side lengths using the trigonometric ratios from the previous slide. Click for the Answer… The first step is to use the trigonometric functions on angle A. Sin θ =6/10 Sin θ =0.6 Csc0.6~36.9 Angle A~36.9 Because all angles add up to 180, B=90-11.537=53.1 The measurements have changed. Find side BA and side AC Sin34=2/BA 0.559=2/BA 0.559BA=2 BA=2/0.559 BA~3.578 The Pythagorean theorem when used in this triangle states that… BC2+AC2=AB2 AC2=AB2-BC2 AC2=12.802-4=8.802 AC=8.8020.5~3

38. B c a A C b When solving oblique triangles, simply using trigonometric functions is not enough. You need… The Law of Sines The Law of Cosines a2=b2+c2-2bc cosA b2=a2+c2-2ac cosB c2=a2+b2-2ab cosC It is useful to memorize these laws. They can be used to solve any triangle if enough measurements are given.

39. B c=6 a=4 28º A C b Solve this triangle Click for answers… Because this triangle has an angle given, we can use the law of sines to solve it. a/sin A = b/sin B = c/sin C and subsitute: 4/sin28º = b/sin B = 6/C. Because we know nothing about b/sin B, lets start with 4/sin28º and use it to solve 6/sin C. Cross-multiply those ratios: 4*sin C = 6*sin 28, divide 4: sin C = (6*sin28)/4. 6*sin28=2.817. Divide that by four: 0.704. This means that sin C=0.704. Find the Csc of 0.704 º. Csc0.704º =44.749. Angle C is about 44.749º. Angle B is about 180-44.749-28=17.251. The last side is b. a/sinA = b/sinB, 4/sin28º = b/sin17.251º, 4*sin17.251=sin28*b, (4*sin17.251)/sin28=b. b~2.53.

40. B c=5.2 a=2.4 A b=3.5 C Solve this triangle: Hint: use the law of cosines Start with the law of cosines because there are no angles given. a2=b2+c2-2bc cosA. Substitute values. 2.42=3.52+5.22-2(3.5)(5.2) cosA, 5.76-12.25-27.04=-2(3.5)(5.2) cos A, 33.53=36.4cosA, 33.53/36.4=cos A, 0.921=cos A, A=67.07. Now for B. b2=a2+c2-2ac cosB, (3.5)2=(2.4)2+(5.2)2-2(2.4)(5.2) cosB, 12.25=5.76+27.04-24.96 cos B. 12.25=5.76+27.04-24.96 cos B, 12.25-5.76-27.04=-24.96 cos B. 20.54/24.96=cos B. 0.823=cos B. B=34.61. C=180-34.61-67.07=78.32.

41. REMEMBER When solving a triangle, you must remember to choose the correct law to solve it with. Whenever possible, the law of sines should be used. Remember that at least one angle measurement must be given in order to use the law of sines. The law of cosines in much more difficult and time consuming method than the law of sines and is harder to memorize. This law, however, is the only way to solve a triangle in which all sides but no angles are given. Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.

42. Part 3 Trigonometric Identities

43. Cofunctions What is the sine of 60º? 0.866. What is the cosine of 30º? 0.866. If you look at the name of cosine, you can actually see that it is the cofunction of the sine (co-sine). The cotangent is the cofunction of the tangent (co-tangent), and the cosecant is the cofunction of the secant (co-secant). Sine60º=Cosine30º Secant60º=Cosecant30º tangent30º=cotangent60º

44. Other useful trigonometric identities The following trigonometric identities are useful to remember. Sin θ=1/cosec θ Cos θ=1/sec θ Tan θ=1/cot θ Cosec θ=1/sin θ Sec θ=1/cos θ Tan θ=1/cot θ (sin θ)2 + (cos θ)2=1 1+(tan θ)2=(sec θ)2 1+(cot θ)2=(cosec θ)2