Quiz 5.1-5.2

denoted by f°g, has the form (f°g)(x)=f(g(x))

• plug x into the second letter function and solve
• plug the solution into the first letter and solve

• 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

• 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

• 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

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

• these always have different y values for unequal x values
• increasing/decreasing functions on an interval I are one-to-one functions on I

• 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)

the reverse of a one-to-one function

• 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

• 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)"

• 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

replace f(x) with y, then interchange the variables x and y; leave in this form

solve the implicit equation for y in terms of x; then have that equal f^-1(x)

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

• 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≠#}

• 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

find the two composite functions and set them equal to each other, then reduce/simplify this expression