Functions Review

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

Evaluating functions

)5()0()3(

12)(

fff

xxf

)3()7()1(

315)(

ggg

xxg

)5()1()0(

4)(

hhh

xxh

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

Find the domain of the following functions:

A)

B)

315)(

x

xg

Domain is all real numbers but 3x

C)

Square root is real only for nonnegative numbers.

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

3210-1-2-3

5

4

3

2

1

0

-1x

y

x

y

Domain and Range from the Graph of a function

Domain = {x / or }

Range = {y / or }

13 x 31 x

21 y 53 y

4

0

-4 (0, -3)

(2, 3)

(4, 0)(10, 0)

(1, 0)

x

y

Determine the domain, range, and intercepts of the following graph.

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.

x

y

Not a function.

x

y

Function.

4

0

-4 (0, -3)

(2, 3)

(4, 0)(10, 0)

(1, 0)

x

y

Is this a graph of a function?

Even functions

A function f is an even function if

for all values of x in the domain of f.

Example: is even because

)()( xfxf

13)( 2 xxf

)(131)(3)( 22 xfxxxf

Odd functions

A function f is an odd function if

for all values of x in the domain of f.

Example: is odd because

)()( xfxf

xxxf 35)(

)()5(5)(5)( 333 xfxxxxxxxf

Determine if the given functions are even or odd

23

3

24

)()4

1||)()3)()2

1)()1

xxxk

xxhxxg

xxxf

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!)

3210-1-2-3

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

x

y

x

y

52.50-2.5-5

5

2.5

0

-2.5

-5

x

y

x

y

Determine if the function is even or odd?

Determine if the function is even or odd?

52.50-2.5-5

5

3.75

2.5

1.25

0

x

y

x

y

52.50-2.5-5

100

50

0

-50

-100

x

y

x

y

Determine if the function is even or odd?

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

52.50-2.5-5

100

50

0

-50

-100

x

y

x

y

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.

52.50-2.5-5

5

2.5

0

-2.5

-5

x

y

x

y

Example of limits at infinity

• The function can converge

The function converges to a single value (1), called the limit of f.

We writelimx + f(x) = 1

Example of limits at infinity

• The function can converge

The function converges to a single value (0), called the limit of f.

We writelimx + f(x) = 0

Example of limits at infinity

• The function can diverge

The function doesn’tconverge to a single value but keeps growing.

It diverges. We can writelimx + f(x) = +

Example of limits at infinity

• The function can diverge

The function doesn’tconverge to a single value but its amplitudekeeps growing.

It diverges.

Example of limits at infinity

• The function may neither converge nor diverge!

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 writelimx + f(x) = +limx - f(x) = 0

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 writelimx + f(x) = 0

Examples of limits at x=0 (x becomes very small!)

• The function can have asymptotes (it diverges). The limit at 0 doesn’t exist…

Examples of limits at x=0

• The function can have a gap! The limit at 0 doesn’t exist…

Examples of limits at x=0

• The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

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

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!)

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…

ac

b

ө A

B

C

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).

Sinθ=

Cos θ=

Tan θ=

Side Opposite

Side Adjacent

Side AdjacentSide Opposite

Hypothenuse

Hypothenuse

=

=

= a

bca

b

c

Try finding the angles of the following triangle from the side lengths using the trigonometric ratios from the previous slide.

610

8

θ A

B

C

α

β

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

C

2

34º A

B

α

β

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

When solving oblique triangles, simply using trigonometric functions is not enough. You need…

The Law of Sines

Cc

Bb

Aa

sinsinsin

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.

a

c

bA

B

C

a=4

c=6

bA

B

C

28º

Solve this triangleClick 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.

a=2.4

c=5.2

b=3.5A

B

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.

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.

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

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

The following trigonometric identities are useful to remember.

(sin θ)2 + (cos θ)2=1

1+(tan θ)2=(sec θ)2

1+(cot θ)2=(cosec θ)2