1.5 Inverse Functions

What is an inverse?

Graphically, an inverse of a function is the reflection of that function over the line y = x. This translates into algebra as a function that undoes another function.

This picture shows the function f(x) reflected in the line y=x to make the inverse function f^-1(x).

Notation:

The inverse of a function f is denoted by f^-1(read f-inverse).

Does every function have an inverse?

No! Not every function has an inverse! In order for a function to have an inverse it must be one-to-one, meaning there is only one y coordinate for every x coordinate.

To find out graphically if a function has an inverse, you must do the horizontal line test.  If a horizontal line crosses the graph at more than one point for any spot that it is placed, then the function does NOT have an inverse.

This function does not have an inverse that exists as a function because it is not one-to-one.











Algebraically, a function does not have an inverse when

A function does have an inverse if and only if,

meaning that no two y-coordinates have the same x-coodinate, unless those coordinates are equal.


How do you find a function's inverse?

To find the inverse of a function, you must switch the x and y coordinates and then solve for the new y.  This will undo everything done in the equation, which by definition will leave you with the inverse function.


Additional Properties of Inverse Functions

Two functions f and g are inverses of each other if

and

Links

A helpful, but long video http://www.youtube.com/watch?v=nSmFzOpxhbY

A shorter helpful video http://www.youtube.com/watch?v=wSiamij_i_k

1.4 Combinations of Functions

Arithmetic Combinations of Functions

Sum: (f + g)(x) = f(x) + g(x)
Difference: (f - g)(x) = f(x) - g(x)
Product: (fg)(x) = f(x) x g(x)
Quotient: (f/g)(x) = f(x)/g(x)

f(x)= 2x+1
g(x) x^2+2x-1

(f+g)(x)=f(x) + g(x)
            =(2x-1) + (x^2+2x-1)
            =x^2 + 4x

(fg)(x)= f(x)g(x)
         = (x^2)(x-3)
         = x^3 -3x^2
(fg)(4)= 4^3- 3(4)^2
         = 16

Compositions of Functions

(fog)(x)= f(g(x))

f(x)= x+2
g(x)= 4-x^2

(fog)(x)=
f(g(x))=
f(4-x^2)=
(4-x^2)+2=
-x^2+6=

A function has an inverse when you plug it into the other one ((fog)(x)) and the input "x" is the same as the output.

1.3 shifting, reflecting, and stretching graphs

Here are the graphs of some of the most commonly used functions in algebra

constant function f(x)= c

identity function f(x)= x

absolute value function f(x)=abs(x)

 square root function f(x)=√x

quadratic function f(x)=x^2

cubic function f(x)=x^3

Horizontal and Vertical Shifts:

let c be a positive real number.  Vertical and horizontal shifts in the graph of y=f(x)

are represented as follows...

vertical shift up h(x)= f(x) + c

vertical shift down h(x)=f (x) - c

horizontal shift right h(x)= f(x-c)

horizontal shift left h(x)= f(x+c)

Reflection in the coordinate axes:

reflection in the x axis h(x)= -f(x)

(this makes all the  y coordinates negative while the x coordinates stay positive, so the graph would be reflected across the x axis)

reflection in the y axis h(x)= f(-x)

(this makes all the x coordinates negative while the y coordinates stay positive, so the graph would be reflected across the y axis)

Stretching/compressing:

if the transformation of the graph y=f(x) is represented by y=cf(x)...

vertical stretch if c>1

vertical compression if 0<c<1

if the transformation is represented by y=f(cx)...

horizontal stretch if 0<c<1

horizontal compression if c>1

I think that basically covers all we need to know since this is all review.

-Megan

Function Definitions

FUNCTIONS 12/6/12

Today we defined some important concepts regarding functions. To begin, remember that a function relates an input to an output, and there can only be one valid output for each input (or only one y value for x value). 

Below are some of the definitions we discussed in class with explanations

A function ƒ is increasing on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies that    

In other words, your smaller x input value will always give you your smaller y output. This will cause the graph of an increasing equation to have a positive slope within your interval.

A function ƒ is decreasing on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies that    

Basically, this means the opposite of the previous definition. In simpler words, it means that for your smaller x input, you will get your larger y output. This would mean that the graph of a decreasing function would have a negative slope on your interval. 

A function ƒ(a) is called a relative maximum of ƒ if there exists an interval (X₁, X₂) that contains a such that 

 implies 

To better understand this definition, you can think of it like f(a) > f(anything else on the interval). On a graph of a relative maximum function like this, you would see a parabola opening downward with a maximum. This maximum would be at the coordinate (a, f(a)). So by inputting the value a, you will get the greatest possible output. This maximum point is also known as the extremum. 

A function ƒ(a) is called a relative minimum of ƒ if there exists an interval (X₁, X₂) that contains a such that 

   implies

An easy way to think about relative minimums is to remember that f(a) < f(anything else on the interval). So, when you input a, you will get your least possible output value f(a). The minimum point on the parabola (which opens upward) is also called the extremum. 

With all of the above definitions, it is important to remember that they only refer to your interval. The graph of a function can change and do crazy things outside of your interval, but you only need to look at the chunk of the graph that's in your interval. 

A function f is even if, for each x in the domain of f,

f (-x) = f (x) 

So basically, when you plug -x into your function, you get the same output that you would get by just plugging in x. By looking at a graph, you can recognize that a function is even. Even functions have symmetry about the y axis. For example, a parabola centered on the y axis would be an even function because each side is reflected over the axis. 

A function f is odd if, for each x in the domain of f, 

f (-x) = - f (x)

In other words, when you plug -x into your function, your output will be the same as -1( f (x) ). You can recognize an odd funciton on a graph because they appear to be reflected about the origin, like the graph below. 

Functions are super important so make sure you understand this stuff. Alright well that's it. Just remember these definitions when you're tryna function. 

 

-Corn 

oh and heres a picture of a young Mr. Wilhelm looking dapper as always. Google works wonders. 

       

Difference Quotients

Difference Quotients!

The purpose of finding a difference quotient is to find the instantaneous slope at a point on a curve. This is done by finding the slope.  In detail:

• Consider two values along the x-axis, x and x+h, where h is the distance between the two points.  The smaller h is, the more exact the difference quotient will be.

• Calculate the slope of the line passing between the points A: (x, f(x)), and B: ((x+h), f(x+h)).

• Simplify

This is the difference quotient, and can be used for any values of x and x+h.

 Here's an Example:

 And then you get to learn limits, but shhh, we won't tell Mr. Bruns...

 This has been just a glimpse at difference quotients... more to come at the end of the year! Yay! but no spoilers...

Blog post by Erin J