Question 114569
#1


If we use the equation {{{D=rt}}}, then solving for t we get {{{t=D/r}}}. Since we know that we're traveling at 58 mph, this means r=58. So we then get {{{t=D/58}}}. Now since we want to find the time based on a given distance, t will be dependent on D (whatever D is, t will be affected by it). So this means D is x (our input) and t is f(x) (our output)


So our equation is:


{{{f(x)=x/58}}}


So if we traveled 58 miles, then we simply plug it in:


{{{f(x)=58/58}}}


And simplify



{{{f(x)=1}}}


Since f(x) is t, the time it takes to travel 58 miles going 58 mph is 1 hour.



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#2



Now to find {{{f^(-1)}}}, simply switch x and f(x) to get



{{{x=f(x)/58}}}



Now solve for f(x)



{{{58x=f(x)}}} Multiply both sides by 58


So our inverse function is {{{f^(-1)(x)=58x}}}



Now heres what happened: When I switched x and f(x), I really switched t and D. So now the distance D is dependent on the time t (whatever t is, D will be derived from t). So for the inverse, x is now t and f(x) is now D. Basically, the inverse now computes the distance given a certain time (instead of a certain computing a certain time given a distance). 


Is this making sense?