What is the significance of composite functions?

Question

How do you evaluate a composite function (f ° g )(x)? Demonstrate with an example of your own or from the text. Explain the concept of inverse of a function y = f(x). Does every function have an inverse? What is the test for determining if a function has an inverse?

Answer 1

What Steve said is spot on. In addition to the horizontal line test there is an algebraic way to determine one to one

You start by assuming 2 arbitrary range values are equal, if we show that their corresponding domain values are equal then we get that the function is one to one.

In others words if a and b were elements in the domain of a function f then we assume f(a) = f(b) and if we arrive at a = b then the function is one to one.

ex. determine if f(x) = x³ + 2 is one to one.

Start with 2 domain elements a and b then f(a) = a³ + 3 and f(b) = b³ + 3 

Then f(a) = f(b) becomes a³ + 3 = b³ + 3. After doing a little bit of algebra you will get that a = b so f(x) is indeed one to one.

This method is helpful if don't know what a graph looks like. If you know what the graph looks like then by all means go with the horizontal line test.

Answer 2

Hi, Joey.

A function composition involves "plugging in" one function into another one. As an example, lets say one function is f(x) = x+1 and another function is g(x) = x^2  We wish to compute the function composition f o g (read f circle g). This means you substitute for x the  g. In other words f o g = f ( g)  you just "plug in"  g into f wherever you see an x. So f(x) = x+1. Now just plug in the x^2 into this:

f o g = f(g) = (x^2) + 1  Notice that the x^2 has taken the place of the x.  Lets try another one. Suppose f(x) = | 6x + 3| and g(x) = 10x - 5  Lets find fog. So f o g = f(g) . Lets now plug the g into the f:

fog = f(g) = | 6(10x-5) + 3| = |60x -27|

You now ask how to find the inverse of a function. The inverse is found simply by interchanging the x and y variable and solving for the y. As an example. Find the inverse of y = 3x - 5. 

step 1:  Swap the x and y:  x = 3y - 5

step 2:  Solve for y:  y = (x+5)/3  = 1/3 x + 5/3  

Not all functions have an inverse. Only those functions that are called one-to-one have an inverse. How do you determine if a function is one to one? If it passes the horizontal line test. If you can draw a horizontal line everywhere through the graph and it intersects in only one place, it is one to one and has an inverse. So if you have a graph that is a parabola like y = x^2, you will notice that the horizontal line crosses in more than one place. Therefore the parabola fails the horizontal line test and is not one to one and does not have an inverse.