Operations with Functions Section 2.4

 Sum  Difference  Product  Quotient  Composition Types of Operations

 Sum: (f+g)(x)=f(x)+g(x)  Difference: (f-g)(x)=f(x)-g(x)  Product: (f*g)(x)=f(x)*g(x)  Quotient: (f/g)(x)=f(x)/g(x) Math Mumbo Jumbo

 Add or subtract like terms  Watch out for negative signs  Watch your parentheses Basically….

 Let f(x)=5x 2 -2x+3 and g(x)=4x 2 +7x-5 Find f +g and f-g Example 1

 Let f(x)=5x 2 -2x+3 and g(x)=4x 2 +7x-5 Find f *g and f/g Example 2

 Try these:  Let f(x)=-7x 2 +12x-2.5 and g(x)=7x 2 -5  Find f+g and f-g  Find g-f Let f(x)=3x 2 +1 and g(x)=5x-2 Find f*g and f/g In your groups

Worksheet

 What on earth does that mean?  When you apply a function rule on the result of another function rule, you compose the functions  In other words, where there is an x in the first function, you actually plug the entire second function in it. Composition of Functions

Example

Example 2

Worksheet

Inverses of Functions Domains and Ranges Horizontal Line Test Relating Composition to Inverses

 Helps us with inverses, by being an easy way to identify them Why learn composition?

 Basically an inverse switches your x and y coordinates  A normal ordered pair reads (x,y) while an inverse reads (y,x) What is an inverse?

 Find the inverse of the relation below  {(1,2), (2,4), (3,6), (4,8)} Example

 The domain of an inverse is the range of the original function  The range of an inverse is the domain of the original  Domain and range flip just as the x and y flip for an inverse. Domain and Range of an inverse

 Find the inverse of the relation below  {(1,2), (2,4), (3,6), (4,8)}  Find the domain of the relation  Find the range of the relation Example

 1. Interchange x and y  2. Solve for y.  Example: y=3x-2 Solving an equation for an inverse

 y=.5x-3 Find the inverse

 Use the horizontal line test  The inverse of a function is a function if and only if every horizontal line intersects the graph of the given function at no more than one point  Look at the original graph,  If it passes the vertical line test, the graph is a function  If it also passes the horizontal line test, the inverse of the graph will also be a function Is the inverse a function?

 If a function has an inverse that is also a function, then the function is one to one

Composition of a function and its inverse

 Show that f(x)=-5x+7 and g(x)=-1/5x+7/5 are inverses of one another

 Graphic organizer Summary