SOLUTION: Can someone show me how to do a problem like this? This is a one to one function f(x)=sqare root of x-3. List the domain and range of the functin and its inverse in internal not

Algebra ->  Functions -> SOLUTION: Can someone show me how to do a problem like this? This is a one to one function f(x)=sqare root of x-3. List the domain and range of the functin and its inverse in internal not      Log On


   

Question 169150: Can someone show me how to do a problem like this?
This is a one to one function f(x)=sqare root of x-3. List the domain and range of the functin and its inverse in internal notation.
Show that f of f^-1(x)=x
f^-1 of f= x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, the domain of the inverse function is simply the range of the original function. Since the range of the original function is y%3E=0, the domain of the inverse function is x%3E=0


Likewise, the range of the inverse function is the domain of the original function. Since the domain of the original function is x%3E=3, this means that the range of the inverse function is y%3E=3


If those two previous statements made no sense whatsoever, then just keep in mind that the domain and range switch when it comes to the inverse function.



Finding the inverse:


f%28x%29=sqrt%28x-3%29 Start with the given function


y=sqrt%28x-3%29 Replace f(x) with y


x=sqrt%28y-3%29 Switch x and y. The goal now is to solve for y


x%5E2=y-3 Square both sides


x%5E2%2B3=y Add 3 to both sides.


So the answer is y=x%5E2%2B3 which means that the inverse function is with the restriction x%3E=0






"Show that f of f^-1(x)=x"


f%28x%29=sqrt%28x-3%29 Start with the given function


f%28f%5E%28-1%29%28x%29%29=sqrt%28x%5E2%2B3-3%29 Plug in . In other words, replace each "x" in f(x) with x%5E2%2B3


f%28f%5E%28-1%29%28x%29%29=sqrt%28x%5E2%29 Combine like terms.


f%28f%5E%28-1%29%28x%29%29=x Take the square root of x%5E2 to get x


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"Show that f^-1 of f(x)=x"


Start with the inverse function


Plug in f%28x%29=sqrt%28x-3%29


Square sqrt%28x-3%29 to get x-3


Combine like terms.