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The document contains class notes on inverse functions from pre-calculus group A. It provides examples of finding the inverse of simple functions algebraically. The notes show that the inverse of f(x)=x/2x-1 is the same function, as it is its own inverse. Another example finds the inverse of f(x)=2√(x-1) to be f^-1(x)=x^2/4+1. The notes also discuss how the domain of a function is the range of its inverse. Finally, it provides one more example of finding the inverse of f(x)=96,000+80x, which is f^-1(x)=1/80x-1200.
Class notes
- 1. Class Notes: 26-11-2012 Pre-Calculus Group A Contributed by Jose Antonio Weymann Inverse Function (Trivial Result, Just Exemplifies) f(x)= x/ 2x-1 x= y/ 2y-1 Switch x’s with y’s x(2y-1)= y isolate y 2xy-x= y 2xy-y-x= 0 2xy-y=x y(2x-1)= x f^-1= x/2x-1 This is the inverse of f(x), in this case the function is its own inverse, this is a rare example and it just serve to show the properties of an inverse function. Now lets prove there are inverses mathematically with f(3) f(3)= 3/6-1= 3/5 f(3/5) (3/5)/(6/5-1)= (3/5)/(1/5) (3/5)*(5/1)= 3/1= 3 Again it proves it is its own inverse. Lets try another problem f(x)= 2 √(x-1) x= 2√(y-1)
- 2. x^2/4 = y-1 x^2/ 4+1 = y f^-1= x^2/4+1 f^-1= 1/4x^2+1 Now let’s evaluate the Domain and Range in functions and their inverses g(x)= x^2 D: all Reals R: [0, oo) The domain of one is the range of the other, so… +/- √(x)= y D: [0, oo) R: all Reals Let’s try another problem f(x)= 96,000+80x x= 96,000+80y x-96,000/80= 80y/ 80 y/80-1200= y f(x)^-1= 1/80x-1200 End of lesson