1.5 Inverse Functions

Reverse order of operations is necessary to to undo/solve a solution.

Exp
Why do we add the 3 to the 7? Its is because it is the inverse operation which is adding adding when the other side of the equation is subtrcting.hy do we add first? it is because we are solving or undoing the equation so we use the reverse order of operations.

Finding an inverse for a solution such as 2x+5=y is simple just switch the x and the y

This is the inverse of the function

Two functions f and g are function iff (if and only if)

f f^-1

(x,y) (y,x)

Only functions that pass horizontel line- Functions have inverses- one to one- for every y theres only one x

A function f is one to one iff:

f(a)=f(b) implies a=b

Use definition to verify

and

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1.4 Composition of Functions

The composition of the function f with g is:

The domain of (f o g) is the set of all x in the domain of g such that g(x) is in the domain of f.

Ex. if

for

 then

 

 

The Domain of (f o g) is [1, infinity).



However, there are times when (f o g) does equal (g o f).

1.4 Combinations of Functions

Arithmetic Combinations-

(f + g) (x) = f (x) + g (x)

Ex. if,

f (x)= 2 - x and g (x)= (x / 4) - 3

Then,

(f + g) (x) = (2 - x) + [(x / 4) -3]

(f - g ) (x) = (2 - x) - [(x / 4) - 3]

(f · g ) (x) = (2 - x) · [(x / 4) - 3]

(f / g ) (x) = (2 - x) / [(x / 4) - 3]

To find the value of (f + g) (x) when x = 0, plug in 0 for the x values

- add the y value for "f (x)" to the y value for "g (x)"

(f + g) (0) = (2 - 0) + [(0/4) - 3]

= (2) + (-3)

= -1

The same process is followed for subtraction, multiplication and division...

(f - g) (0) = ( 2 - 0) - [(0/4) - 3]

= (2) - (-3)

= 5

(f · g) (0) = (2 - 0) · [(0/4) - 3]

= (2) · (-3)

= -6

(f / g) (0) = (2 - 0) / [(0/4) - 3]

= (2) / (-3)

= (-2/3)

1.3 Transforming Function

Today we learned about transforming functions,translating, reflecting, and stretching/compressing.

Here is an original problem:

y = f (x)

Translating Functions
Translation occurs when we move the graph units up/down or right/left

Moving a function up or down:

y = f (x) + c 

where c is the number of units moved.  If c is positive (as it is above), the function will move up c units; however, if c is negative, the function will move down c units.

If we were to translate the original problem down two units, the end formula would look like:

y = f (x) - 2

Moving a function left or right:

y = f (x - c)

C is still the number of units moved, however the graph is now moved to the left or to the right.  If c is positive, the function will move c units to the left.  If c is negative, the function will move c units to the right.

If we were to translate the original problem to the left 4 units, the end formula would look like:

y = f (x + 4)

A combination of movements left/right and up/down would look like this:

y = f (x - 3) + 2

where the original function will be translated 3 units to the right and up 2 units.

Stretching or compressing functions

Stretching/Compressing occurs when we move the graph c times away from the x or y axis.

Stretching a function:

y = c * f (x)

where c is the number of units moved.  The graph will be pulled from the x-axis c times.  The  x-intercepts stay the same.

If we were to stretch the original problem by 2, the end formula would look like:

y = 2 f (x)

which would result in a 2x distance from the x-axis.

Compressing a function:

y = f (c * x)

C is still the number of units moved, however the graph is now compressed.  If c is a large number (like 42), the function compresses more closely to the y-axis.  If c is a small number (like 0.2), the function stretches from the y-axis.

If we were to compress the original problem by 4, the end formula would look like:

y = f (4x)

which would result in a 4x distance from the y-axis.

A combination of stretching from the x and y axis would look like this:

y = 3 f (0.1 x)

where the original function will be 3x away from the x-axis and 0.1x away from the y-axis [which would actually be closer because it is less than the original function (which was 1)].

Reflecting Functions

Reflecting occurs when we reflect the graph across the x or y axis.

Reflecting a graph across the x-axis looks like:

y = - f (x)

where the function's x-coordinates remain the same; however, all the y-coordinates become negative, and reflect across the x-axis.

Reflecting a graph across the y-axis:

y = f (-x)

where the function 's x-coordinates become negative; however, all the y-coordinates remain the same, and reflect across the y-axis.

Combination of the Functions

A combination of all three types of transforming functions would look like:

y = -3 f (x + 2) - 8

where the graph is:

• reflected across the x-axis
• stretched 3 units away from the x-axis
• translated 2 units to the left
• and translated 8 units down.

Even and Odd Funtions

Here are some even numbers:
2, 4, 6

Here are some odd numbers:
1, 3, 5

Here are some that aren't even or odd:



So how do we determine if a function is even or odd?


A function f is even if, for each x in the domain of f, f(-x) = f(x)
f(-x) = f(x)

Example:

f(x) = - 7
f(x) = (-) - 7
- 7 even (same as the original function)

A function f is odd if, for each x in the domain of f, f(-x) = -f(x)
f(-x) = -f(x)

Example:
f(x) = - 4x
f(-x) = (-x)^3-4(-x)

=-x^3 + 4x

Then prove it's odd
-f(x) = -1 (x^3 - 4x)
= -x^3 + 4X
(same as f (-x))

9/16 - Difference Quotient

This is the difference quotient. An example of how to use the equation is as follows:

f(x) = 2x - 1

Plug it into the difference quotient:

And solve simplify:

 

 

Other notes:

REMEMBER: the Domain is the set of possible inputs. The Range is the set of possible outputs.

Consider the equation  function f(x)=2x-5

• The Domain on this equation function is (-∞,∞), because you can plug any number into the equation to yield a useable result that makes sense with the graph.

Now consider the equation function

• The domain for this equation function is (-∞,3)U(3,∞). The graph cannot touch three, because the denominator cannot equal zero.

Lastly, lets look at the equation function

• This equation function has a domain of [5,∞). the input has to be greater than or equal to five because you cannot take the square root of a negative number.

9/15/10 (Chapter 1.1: Functions)

Chapter 1.1 - Functions:
... know that f(x) = y

• A function is a relation that assigns exactly one value in the domain to each value in the range. This means that for every x-value, there is only one y-value.

• The domain is...
  - the x-values
  - the input
  - the independent variable
• The range is...
  - the y-values
  - the output
  - the dependent variable
  
  
  

How to tell if an equation or a graph is a function:

• conduct a vertical line test...
  --> if the vertical line only intersects the graphed equation in one spot, it is a function.
  
  
  
  
  --> if the vertical line intersects the graphed equation multiple times, it is not considered a function.

• solve for y...
  • this is a function, the graphing of this equation will result in a parabola opening upwards.
    .
    
    
  • this is not a function, the graphing of this equation will result in a parabola opening to the right. (this would not pass the vertical line test)
    
    

!!! Indications that an equation might not be a function:

• if the y-value is in an absolute value (there would be two y-values for one x)
• if the y-value has a power greater than 1.