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Operations with Functions

Operations with Functions. Section 2.4. Types of Operations. Sum Difference Product Quotient Composition. Math Mumbo Jumbo. 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). Basically…. Add or subtract like terms

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Operations with Functions

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  1. Operations with Functions Section 2.4

  2. Types of Operations Sum Difference Product Quotient Composition

  3. Math Mumbo Jumbo 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)

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

  5. Example 1 Let f(x)=5x2 -2x+3 and g(x)=4x2 +7x-5 Find f +g and f-g

  6. Example 2 Let f(x)=5x2 -2x+3 and g(x)=4x2 +7x-5 Find f *g and f/g

  7. In your groups • 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-2 Find f*g and f/g

  8. Worksheet

  9. Composition of Functions 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.

  10. Do not confuse the symbol for composition, open dot, with the symbol for multiplication (closed dot) f

  11. Example Let Find f

  12. Example 2 Let and g(x)=3x+1 Find f

  13. Worksheet

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

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

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

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

  18. Domain and Range of an inverse 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.

  19. Example 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

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

  21. Find the inverse y=.5x-3

  22. Is the inverse a function? • 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

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

  24. Composition of a function and its inverse 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

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

  26. Summary Graphic organizer

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