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Gymnastxoxo17
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composite function form
denoted by f°g, has the form (f°g)(x)=f(g(x))
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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
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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
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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
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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
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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
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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
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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)
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inverse function f-1(x)
the reverse of a one-to-one function
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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
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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)"
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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
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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
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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)
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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
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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≠#}
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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
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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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