Blog 4: Composition of Functions with Anna Monastero

Hi everyone! Today we'll be learning about the composition of functions. So, first off, what is a function?

A function is the relationship between an input and an output. Most commonly, functions are written in the form of f(x)=___ , where x is the input and what f(x) is equal to is the output.

This means that by changing the value of the input, we can subsequently change the value of the output. For example:

By making the input 4, we are saying that 4 will be substituted for x. This occurs on both sides of the equation and results in the above equation. Also, remember that the x value is just a placeholder. It can take the value of any number imaginable and will result in an output. However, if this is not the case and an x value does not get you  y value(output), then you do not have a function.

Now that we know the basics of a function, we can compose them. 

In the problem above we are given two functions. Our goal is to get an equation for the answer of f(g(x)) and g(f(x)). This is much simpler than it looks in the example and only requires a few steps. 

     1. Figure out your "inside" and "outside" functions. The outermost function in this case would be f(x) and your innermost function is g(x).

     2. Next, set up the function. 3x^2+12x-1 is your starting point because it is your outermost function

     3. Now you need to plug the g(x) value into the f(x) equation. Therefore, for every x in the f(x) function, plug in g(x). It should look like 3(4x+1)^2 +12(4x+1)-1

     4. The next step is the easy part, just solve the equation!

     5. Lastly, f(g(x)) should equal 48x^2+72x+14

Overall, functions can be used in numerous ways in everyday life. If you buy something at the grocery store, you have a function. The input of a certain amount of money will get you a specific output. This means that if you buy a 2$ bag of chips, that 2$ was your input and the bag of chips was your output. In more complex terms, the composition of functions can be used when there are multiple variables to a problem. For example, if you work at a retail store and make commission on top of your hourly pay, this can be calculated by using function composition. Ultimately, functions are not only used in math but are applicable to our everyday life.