Composition: How to prove it is an inverse

If you want to prove that one function is inverse to another, you need to do a composition.

Note: remember f(x) is the original function and f -1 (x) is the inverse function.

1.- You have two options:

Case 1- Instead of writing f-1(x) , substitute x for f(x) in the inverse function

f- -1 (f(x)) =x

Case 2- Instead of writing f(x), substitute x for f- -1(x) in the original function

f (f -1(x)) =x

Example:

Case 1

f(x)=3x-1 and the inverse is f^-1(x)=sqr(x+2)

With the composition it will look something like this:

f (f -1(x)) =3(sqr(x+2)-1

Case 2

It will look like this:

f- -1 (f(x)) =sqr((3x-1)+2)

How to graph a function and its inverse? (Step-by-Step)

​

                                        ​​

​

The first thing you need to know is that the function and its inverse will be symmetric around the line y=x.

When we have the form

y=mx+b

1.- You need first to find the y-intercept which is going to be the independent value or b.

2.- Next we are going to use the rise/run. If you have a function with a value of ¾x, your rise will be of 3 and your run of 4. So you are going to move 3 units in the y-axis and then I move 4 units in the x-axis. And then graph that point

3. Now you're going to have two points that you will need to join forming a line

4. It's time for the inverse, we will do the same but now with the equation of the inverse.

5. Graph the line (try to use another color so that you can distinguish them)

6. Let's graph our line y=x, also called identity line. That will be a dash line where it's y-intercept is zero and has a slope of 1/1

7. If they seem they're reflected to each other, you do it right!

Easy right? Now let's learn how to graph a parabola and its inverse

Now the function will be expressed as something like this

y=ax^2+c

1.- First we need to find the vertex. When x is alone the value of it will be zero, the same happens with the y, if you don't have a constant then its value will be zero.

Important: is there is no negative sign before the a, the parabola will be opening upwards and if it does have a negative sign it will be opening downwards.

2.- Then you will substitute with any number you want in x. Example:

y=-x^2-2

And you decide to use -1 what you will do is the following. (-1)^2=1, in this case our vertex will be at (0,-2) and now you substitute in the -2. Now join the two dots and draw the parabola

3.- Draw your identity line at y=x

4. Now imagine you're placing a mirror in the identity line, how does the parabola look like? Now draw it in the graph

How to convert a function from y=ax2+bx+c to y=a(x - h)2 +k by COMPLETING SQUARES

​

                                               ​​

​

Example: y= x2 -8x+7=0

1.- First, you need to make f(x) (the function/y) equal to 0.

Then, pass the “=0” to the right to make the clearing easier.

x2 -8x+7=0

2.- As a second step, you need to pass the constant term to the other side. (The one without variable)

x2 -8x=-7

3.- Now, we are going to COMPLETE SQUARES.

For this, you need to divide the second term over 2 and elevate it to the square. Afterwards, you have to add the result on both sides of the function.

x2 -8x+ (-8/2) 2 =-7+ (-8/2) 2

x2 -8x+ (64/4)  =-7+(64/4)

x2 -8x+ (16)  =-7+(16)

4.- When having this result, you will need to convert it to (a+b) 2

For doing this, you get the square root of the third term and add the result to the square root of the first term with the sign of the second term. Then, you need to elevate everything to the square.

(x-4) 2 =-7+(16)

(x-4) 2 =9

5.- Equal everything to 0 passing the constant number to the left of the equal sign.

(x-4) 2 -9=0

6.- Make 0 “f(x)” or “y” again and pass it to the left.

f(x)=(x-4) 2 -9

Note: “f(x)” and “y” are the same.

You converted it!

How to get the inverse function

​          

                                        ​​

​​

​1.- For getting the inverse of a function, you ALWAYS need it to be in this form:

y=a (x - h)2 +k

Example:

y=(x-4) 2 -9

In case it is expressed in this form:

y=ax2 +bx+c

y= x2 -8x+7=0

You need to convert it.

(If you don’t know how to convert it, please go to section 3)

2.- First, you need to substitute x per y and y per x.

y=(x-4) 2 -9

x= (y-4)2-9

Then, you just need to isolate for y.

(y-4)2-9=x

(y-4)2=x+9

y-4= -+ √ x+9

y= 4 -+ √ x+9

You got the inverse function!

The inverse function

The Inverse Function

Inverse functions: the first topic you are going to see in third semester. Welcome to “Math for all”, the blog in which the objective is for you to understand every concept and to learn how to get inverse functions.

 

Here are some definitions that might be useful for you….

Relations

Set of ordered pairs. The first elements in the ordered pairs (the x-values), form the domain. The second elements in the ordered pairs (the y-values), form the range.

It can have two different values for the  output.

Functions

Relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

One to One Function

A one to one function is the function that will have only one output per each input, meaning that two different inputs can't produce the same output

Variable

An element, feature, or factor that can vary depending on what we have as values.

Existing variables in the function:

•   Dependent (y)

The dependent variable is the one that will be changing if the independent variable has different values. For example, if I have a one marshmallow and heat it, the dependent variable will be the volume of the marshmallow because it will change depending on the temperature, but the one that will stay constant is the quantity of marshmallows I have.

• Independent (x)

The independent variable is the one that its value will not change, referring that once its value is specified will remain constant throughout the problem

F(x)= Function

F(x)-1 = Inverse function