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Operations with FunctionsSection 2.4
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Sum Difference Product Quotient Composition
Types of Operations
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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
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Add or subtract like terms Watch out for negative signs Watch your parentheses
Basically….
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Let f(x)=5x2 -2x+3 and g(x)=4x2 +7x-5
Find f +g and f-g
Example 1
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Let f(x)=5x2 -2x+3 and g(x)=4x2 +7x-5
Find f *g and f/g
Example 2
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Try these: Let f(x)=-7x2+12x-2.5 and g(x)=7x2-
5 Find f+g and f-g Find g-f
Let f(x)=3x2+1 and g(x)=5x-2Find f*g and f/g
In your groups
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Worksheet
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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
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Do not confuse the symbol for composition, open dot, with the symbol for multiplication (closed dot)
f
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Let Find f
Example
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Let and g(x)=3x+1Find f
Example 2
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Worksheet
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Inverses of FunctionsDomains and RangesHorizontal Line Test
Relating Composition to Inverses
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Helps us with inverses, by being an easy way to identify them
Why learn composition?
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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?
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Find the inverse of the relation below {(1,2), (2,4), (3,6), (4,8)}
Example
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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
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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
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1. Interchange x and y 2. Solve for y. Example: y=3x-2
Solving an equation for an inverse
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y=.5x-3
Find the inverse
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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?
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If a function has an inverse that is also a function, then the function is one to one
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If f and g are functions and (f
Example: Show that f(x)=7x-2 and g(x)=1/7x+2/7 are inverses of one another
Composition of a function and its inverse
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Show that f(x)=-5x+7 and g(x)=-1/5x+7/5 are inverses of one another
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Graphic organizer
Summary