Quiz 5.1-5.2

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Gymnastxoxo17
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268449
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Quiz 5.1-5.2
Updated:
2014-03-30 10:40:48
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  1. composite function form
    denoted by f°g, has the form (f°g)(x)=f(g(x))
  2. how to evaluate a composite function like (f°g)(x)
    • plug x into the second letter function and solve
    • plug the solution into the first letter and solve
  3. how to find the domain of a composite function
    • find the domain of the second letter; restrictions of this apply to the domain of the composite
    • find the domain of the first letter. Find the #s for which, if plugged in to the 2nd letter, would include any of the the #s excluded from this domain.
    • set these restrictions equal to the second letter function, solutions of this need to be excluded from the domain of the composite function
  4. how to show that the two composite functions are equal
    • solve (f°g)(x), you should get x; then solve (g°f)(x), you should get x
    • just plug the second letter function into the first letter function and solve
  5. how to tell if a function is one-to-one based off its domain and range
    • you can't have the same y for different x's 
    • any two different inputs in the domain correspond to two different outputs
    • y-values do not repeat for different x-values
  6. horizontal line test
    if every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one
  7. increasing/decreasing functions
    • these always have different y values for unequal x values
    • increasing/decreasing functions on an interval I are one-to-one functions on I
  8. how to find an inverse function and its domain and range based off a function's domain and range
    • the set of domain values for a one-to-one function equals the set of range values for its inverse function, and vice versa
    • if the function f is a set of ordered pairs (x,y) then the inverse of f is the set of ordered pairs (y,x)
  9. inverse function f-1(x)
    the reverse of a one-to-one function
  10. how to find the domain of a function or its inverse based off the function expression
    • plug the function into its reverse function and solve for the domain of the function; plug the inverse function into the function f and solve for the domain of the inverse
    • f-1(f(x))=x, where x is the domain of f; f(f-1(x))=x, where x is the domain of f-1
  11. how to find the values of x for which (f(f-1(x))=x
    • find the exclusions from the domain of the function being plugged in
    • after you show that the inverse plugged into the function=x, say "provided x≠(those restrictions)"
  12. how to find the graph of an inverse function from the graph of a function
    • whenever (a,b) is on the graph of f then (b,a) is on the graph of its inverse; switch the coordinates of the function to get the coordinates of its inverse
    • the graph y=x is a line of symmetry btwn the points; reflect the graph of the function over this line to get the graph of its inverse
  13. how to find the inverse of a function implicitly based off the function expression
    replace f(x) with y, then interchange the variables x and y; leave in this form
  14. how to find the inverse of a function explicitly based off the function expression
    solve the implicit equation for y in terms of x; then have that equal f-1(x)
  15. how to check your answer after finding the inverse function explicity
    check that you get x when you plug the inverse into its function; or that you get x when you plug the function into its inverse
  16. how to find the domain and range of a function
    • find the domain of it like normal
    • find its inverse function; the range of the function is equal to the domain of that
    • remember express the range in terms of y; R {y|y≠#}
  17. how to find the inverse of a domain-restricted function
    • sometimes you have something that is not one-to-one, but is when you restrict its domain
    • when finding its inverse in implicit form, interchange the variables as well as the restriction
    • when finding its inverse in explicit form, keep the restrictions for the function's domain in mind as you find this; sometimes there will be no restrictions on the domain of the inverse function as well
  18. how to find the conditions for which (f°g)(x)=(g°f)(x)
    find the two composite functions and set them equal to each other, then reduce/simplify this expression

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