Chapter 1

                                                                     Chapter 1 

Book's definition of a Function
               - A set of relation that matches each item from one set with exactly one item from a different set.


                                                              Even Functions definition  

How can you recognize whether or not a function is even based on its graph?

-The graph is symmetrical about the Y-axis

Example: 

This Graph is an Even function because it reflexes on the Y-axis  

How to decide whether if an equation is and even function?

Example:

We want to know this the equation above is an even function. you know the definition on and even function is 

so now you would put the input of the -x in the x position 

 And now salve from here

Now that F(-x) = F(x) you know its an even function 

Odd Function definition 

How can you recognize whether or not a function is odd based on its graph?

- Its symmetrical about the origin 

Example:

This graph is and odd function because it reflexes off the origin  

How to tell if an equation is an odd function?

Example: 

    

 Simplify                                                                                                                 Simplify

                                                                                                      

                       The Two equation equal each other and there for the equation is and odd function  

ONE-TO-ONE definition 

If......Then

How can you recognize whether or not a function is one-to-one based on its graph?

- it passes the horizontal line test 

This graph is a one-to-one function because it passes the horizontal line test.

How to tell if an equation is an one-to-one function?

Example: 

Now set the equation above equal to each other and plug in a on one said and b on the other said and see if they a become equal 

Now simplify

because a and b are equal to each other show that this equation is a one-to-one function 

Inverse Function definition 

If F&G are inverse, then (Fog)(x) = x

What is the relationship between the graph of a function and the graph of its inverse?

- there reflex across each other

The kinda function that has and inverse is a one-to-one function. 

Example: 

Fine the inverse of

Now change the F(x) top y 

Now change the x and the y and solve for y

And this is the inverse!!

Inverse Trigonometric Functions

The book's definition of an inverse trigonometric function is:
y = arcTRIG x  if and only if  TRIG y = x
With the domain and range being restricted
Evaluating

And because it is within the domain we can use the Unit Circle to help solve.

 Negative pi over six is or answer because the sine of pi over six is equal to 1/2 therfore the inverse sine of 1/2 is equal to pi over 6.


Graphing

 These are the normal sine, cosine, and tangent

sin y = x

 

 

cos y = x

 

 





 tan y = x

Now look at the inverse graphs of both and you can see what the difference is.

y = arcsin x

 

Note :

1.  That the graph is reflected in the line y = x
2. It sill passes through (0,0)
3. The Domain and Range have changed

 

y = arccos x

 

 

Note:



1. This graph is also reflected in the line y = x
2. It still starts at 1 just on the x axis instead of the y
3. The Domain and Range have also changed

y = arctan x

Note:

1. It is reflected in the line y = x
2. It has horizontal asymptotes at pi over 2
3. It's domain is now all real numbers

Evaluating Compositions of Functions

If you have an equation like:

 

 

What you can do is basically cancel out the tangent and inverse tangent so that your answer is just.

 

 

However when you don't have the same Trionometric function in the parenthesis. For instance:

 

 



Then you make the trig function in the parenthesis (in this case arcsin 7/4) and you set that equal to theta and make a right trianlge.

 

You make a right triangle with the two sides given then use Pythagorean Theorn to get the third side.

And you end up with the square root of 33 as your answer.

 

You then finish by solving using tangent.

 

 

And There Is Your Answer !!!



 

Graphing the Other Four Trigonometric Functions

Just like the graphs of the sin x and cos x, the graphs of the other four functions have very distinct patterns they can be recognized by. In this blog post we'll be covering the graphs of tan x, cot x, csc, and sec x

These functions also come in the formula        f(x) = a trig[b(x-c)] + d  

The tan x Graph:

To graph the function of tan x, we look back at graphing rational functions to make it easier on us. 

Since tan x equals sin x divided by cos x, we can use that instead. 

By taking each value for the angles in the unit circle and plugging them into the new function, we can find plot-able values for tan x.

When we plot those points, it creates a graph that looks like this. Some important things to remember about the parent tan x graph are:

1. x-intercept at (0,0)

2. Values that make cos x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make sin x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of tan x is pi

Like sin x and cos x, the value of a vertically stretches and compresses the graph but because the graph approaches infinity when it approaches the asymptotes, you can only notice the difference a makes when compared to another tan x graph. If it is negative however, it will be reflected across the x-axis. 

The value of b affects the period of the graph. If it is negative, it will reflect the graph across the y-axis. This looks exactly like reflecting it across the y-axis however. Unlike sin x and cos x, the formula for finding 

the period or b is: 

 Finally, the values of c and d just shift the graph either to the left or right, or up and down, respectively. Remember that the x within the parentheses needs to have a coefficient of 1. Also within the parentheses, positive means left, negative means right. 

The cot x Graph:

The cot x graph works just like the tan x graph.

So by looking at the table towards the top, we just take the values for cos x and divide them by the values for sin x. This gets us the values for cot x.

Plotting the points gets us the graph:

Important things about the parent cot x graph:

1. x = 0 is a vertical asmyptote 

2. Values that make sin x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make cos x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of cot x is pi

The values of a, b, c, and d all work the same way they do with the tan x graph. It also uses the formula  

The csc x Graph:

We also make csc x look more like a rational function by doing:

Looking at the table at the top of the graph, we just divide 1 by the values we see for sin x, then we plot. This gets us a graph that looks like this: 

The graphs of csc x and sec x look a little bit different than x and cot x. These graphs make parabola looking things that are boxed in by vertical asymptotes. 

Important things to note about the parent csc x graph:

1. x = 0 is a vertical asymptote 

2. The values of that make sin x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

Changing the value of a will change how far away the minimums and maximums are from the "middle line" between. To find the "middle line" of the graph, just add the y values of a minimum and maximum together and divide by 2. If it is negative, the graph reflects across the x-axis. 

Changing the value of b affects the period of the graph. Like the sin x graph, the period for the graph of csc is 2 pi. To find the period, use the formula:

Changing the the value of c shifts the graph to the left or right, but pay attention to the sign of your a and b values so you catch which way the graph is actually moves. 

Changing the value of d shifts the graph up or down. In the case of csc x, it is going to shift the "middle line" up or down. 

The sec x Graph:

Same process, turn sec x into a rational function. So, 

Looking at the table above, we find all the values for sec x and then we plot.

That gets us this graph: 

Important things to note about the parent graph of sec x:

1. x = 0 goes right down the center of the the parabola-like shapes 

2. The values of that make cos x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

And finally, the values of a, b, c, and d affect the graph in the same way as they do in the graph of csc x.

So that's nearly everything you need to know about graphing the other four trigonometric functions. Hope this helps!