Finding the Inverse FunctionDate: 04/10/97 at 20:55:44 From: merri brennecke Subject: Finding the Inverse Function Find the inverse function of f(x)=4x/(x+2), where x does not = -2. State the domain and range of each. I am unable to solve for x. The answer I get is y = 2. When I solve for x, they cancel each other out. Thank you for your help. Merri Date: 04/11/97 at 20:34:26 From: Doctor Wallace Subject: Re: Finding the Inverse Function Hi Merri! Recall that inverse functions are symmetrical about the line y = x. Since this is true, the way I was taught to find an inverse was to swap the x and y variables and solve for y. This is how it works in your problem: Begin with 4x 4y y = ----- and swap to get x = ----- . x+2 y+2 Now we solve for y. First we get rid of the denominator on the right by multiplying by y+2. This gives: x(y+2) = 4y. Now, use the distributive property on the left, to get xy + 2x = 4y. Group the y's on the right, to get 2x = 4y - xy. Factor the right side: 2x = y(4-x). And finally divide both sides by (4-x): 2x ----- = y. (4-x) And so there's the inverse! You have to check to make sure that it's a function, and not just a relation. We have to restrict the domain here, as well, or we get a division by zero, so we say x not equal to 4. For the domain and the range, we have to keep the restrictions. Since the original could not have -2 in the domain (x values), the inverse cannot have -2 in the range (y values). Sure enough, if you plug a -2 in for y in the inverse, you get -8 = 0 when you simplify it. I hope this helps. Thanks for writing! Don't hesitate to write again if you have more questions. -Doctor Wallace, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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