Inverse of Functions
Objective
MOVING THE GRAPH UP
Let’s look at a quick x-y chart to see what this means.
X | Y |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
| |
| |
| |
| |
| |
X Y
0
3
1
2
3
4
7
12
So what’s the graph look like?
MOVING THE GRAPH DOWN
X | Y |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
X | Y |
| |
| |
| |
| |
0
-3
1
2
3
-2
1
6
SO WHAT OTHER WAYS CAN WE MOVE THE GRAPH?
So moving the graph up and down was pretty simple, but what about moving the graph left or right?
This gets a little more complicated, so let’s experiment.
First off, we know that y goes up and down, so adding or subtracting something from y is pretty easy.
However, x is the variable that determines left to right, and we can’t just add something to the equation because that’s how we move up and down.
So, any ideas?
Well, let’s just try a few things and see what happens
MOVING THE GRAPH HORIZONTALLY
MOVING THE GRAPH LEFT
X | Y |
| |
| |
| |
| |
-3
0
-2
-1
0
1
4
9
That was different.
So then, what’s the graph look like?
So, why did we pick -3 as our first point and not 0?
MOVING THE GRAPH TO THE RIGHT
X | Y |
| |
| |
| |
| |
3
0
2
1
0
1
4
9
It makes sense, so what does the graph look like?
So as we can see, it did shift to the right 3 slots.
But that’s weird, why is it that subtracting from x moves the graph to the right,
but adding moves it to the left?
It almost seems like it’s an inverse operation.
THE RULE FOR TRANSFORMING GRAPHS
SO, HOW DO WE MAKE THE GRAPH STRETCH OR SHRINK THEN?
There are actually a few more things we can do to the graph to change it (without actually changing the graph completely).
We can stretch the parabola, and we can constrict the parabola.
Stretching and constricting (shrinking) the parabola is a little harder to see than just moving it, so explaining how it works is going to be a little tougher.
It’ll be easier if we just dive in.
SHRINKING A PARABOLA
X | Y |
| |
| |
| |
| |
0
0
1
2
3
2
8
18
So our graph changes from:
To:
So how do we make it stretch?
STRETCHING A PARABOLA
X | Y |
| |
| |
| |
| |
0
0
2
4
6
2
8
18
Again, we went from this:
To this:
FINDING THE INVERSE OF A FUNCTION
To find the inverse of a function, first we need to know what the inverse of a function is.
So, in mathematics, an inverse function is a function that undoes the action of another function.
In other words, an inverse function could be considered the opposite function of a given function.
To find the inverse function, we have a few rules that we need to follow, but once found, it can help us better understand the function we are working with.
Again, this is more confusing to explain than it is to show, so here’s what I mean.
Find the inverse of the function
To find the inverse of a function, first we need to start with the original function.
So, something like:
Since we have only 1 variable (x), let’s change f(x) to y for simplicity’s sake.
To find the inverse, the first thing we are going to do, is solve for x.
However, it’s important to note that we’re not going to find a number for x.
Instead, we’re going to find an expression that is equal to x.
So, let’s begin.
So first, we need to get rid of the fraction by multiplying both sides by 4:
Now, let’s add 2 to both sides:
Finally, we divide both sides by 3, and we are left with:
___________
3 3
Lastly, we switch our x’s and y’s, and we have our inverse function:
Now we have our inverse function!
The way we would label this, is if:
How do we check the inverse function?
We put the inverse function inside the original function.
Now, it’s been a long time since you have done this (or maybe you never have), so just in case, let’s review over function notation:
Function notation (a quick review)
So far, we’ve seen that a function usually goes by the name: f(x),
which is pronounced “f of x”.
To start off with, let’s have a real function:
Now, when you change the x in f(x), what that means is wherever you see
An x in the equation, you replace it with whatever is in the parenthesis.
So, for example:
However, we don’t always need to use numbers.
Sometimes, we’ll use another variable, such as:
But mostly, we use functions so we can combine functions.
So, for example, let’s say we have another function:
Now, what we’re going to do is combine these two (pronounced “g of f of x”:
Since f(x) = 3x, then wherever we see an x in g(x), we’re going to put 3x like so:
So, g of f of x is equal to: 12x – 2
Now, it’s very important to note that g(f(x)) may not always be equal to f(g(x))
For example:
Back to the inverse functions
So how do we know if the function we found is the inverse function?
We plug it in and see if we get x as an answer.
For example, we know that:
So we got x as our final reduction!
Example 2
Find the inverse function if f(x) =
Again, we have only 1 variable (x), let’s change f(x) to y for simplicity’s sake.
To find the inverse, the first thing we are going to do, is solve for x.
However, it’s important to note that we’re not going to find a number for x.
Instead, we’re going to find an expression that is equal to x.
So, let’s begin.
So first, we need to get rid of the fraction by multiplying both sides by 5:
Now, let’s add 3 to both sides:
Lastly, we switch our x’s and y’s, and we have our inverse function:
Now we have our inverse function!
The way we would label this, is if:
Example 3
Find the inverse function if f(x) =
Again, we have only 1 variable (x), let’s change f(x) to y for simplicity’s sake.
To find the inverse, the first thing we are going to do, is solve for x.
However, it’s important to note that we’re not going to find a number for x.
Instead, we’re going to find an expression that is equal to x.
So, let’s begin.
So first, let’s subtract 9 from both sides:
Now, let’s multiply both sides by 4:
Finally, we take the square root of both sides, and we are left with:
Lastly, we switch our x’s and y’s, and we have our inverse function:
Now we have our inverse function!
The way we would label this, is if: