Function Compositions and Inverses

Function Compositions and Inverses

Some text here about the topic of discussion

Let's learn how to compose functions and create function inverses

​Learning Target

​Function Composition

Function Composition is just more substitution, very similar to what we have been doing with finding the value of a function.  The difference is we will be substituting another function instead of a number ...

​Function Composition

Given f(x) = x - 5, find f(a+1)

Substitute (a+1) for x

f(a + 1) = (a + 1) - 5

             = a+1 - 5

             = a - 4

​Function Composition

• Composition notation looks like g(f(x)) or f(g(x)), we read this f of g of x’. 

•We are given f(x) and g(x), the function inside the parentheses gets substituted into the other.

Open Ended

Given the functions: f(x) = 2x+2 & g(x) = 2

                find f(g(x))

Open Ended

Given the functions: g(x) = x - 5 & f(x) = x + 1

      

find f(g(x))

Open Ended

Given the functions: f(x) = x2 + x & g(x) = x - 4

              

find f(g(x)) and g(f(x))

      

Open Ended

Given Given  f(x) = 2x + 5  & g(x) = 8 + x

find f(g(-5) & g(f(-5)

              

      

​Function Inverse

Remember:  a function is a set of ordered pairs (including lists of discrete points and also equations which give us infinite points), where no two points have the same x-coordinate.

The Inverse of a function is the set of points where each point in the function is reversed, (y, x).

​Function Inverse

A function that is a list of ordered pairs is easy to find the inverse of:

  f(x) = {(1, 2), (2, 5), (3, -4), (4, 0)}

The inverse is:

  f^-1(x) = {(2, 1), (5, 2), (-4, 3), (0, 4)}

​Function Inverse

To find the inverse of a function that is written as an equation, like:

           f(x) = x + 7

We will:

  1)  Replace the function label, f(x) with y

  2)  Swap the variables, x and y

  3)  Solve the new equation for y

Open Ended

Find the Inverse of:

    f(x) = x + 7

Open Ended

Find the Inverse of:

    f(x) = 3x - 4

Open Ended

Find the Inverse of:

     f(x) = (2x + 5)/3

Let's learn how to compose functions and create function inverses

​Learning Target

Open Ended

Find the Inverse of:

      f(x) = x² - 4