Quiz 5.15.2
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composite function form
denoted by f°g, has the form (f°g)(x)=f(g(x))

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

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

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

how to tell if a function is onetoone 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
 yvalues do not repeat for different xvalues

horizontal line test
if every horizontal line intersects the graph of a function f in at most one point, then f is onetoone

increasing/decreasing functions
 these always have different y values for unequal x values
 increasing/decreasing functions on an interval I are onetoone functions on I

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 onetoone 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)

inverse function f^{1}(x)
the reverse of a onetoone function

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}

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)"

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

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

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)^{}

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

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 {yy≠#}

how to find the inverse of a domainrestricted function
 sometimes you have something that is not onetoone, 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

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