I want to know how to find the range of the equation y=3/(x^2+1) and y= (x-4)/x without using the graphical method?
Detailed explanation would be appreciated.
Please and thank you.
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This is a linear function. Linear functions have domain and range unless the domain is restricted.
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It should be clear that the denominator is never negative. In fact, the denominator is always .
You should know that as a denominator gets larger, the fraction gets smaller. So that means that the largest possible value can take is , and the fraction will get closer to as increases/decreases.
So the range is .
If I were to rewrite y=1-(4/x) in the format of y=(a/x), it would be y=-(4/x)+1. What does the negative in front of (4/x) do to the graph or what does it mean? Our teacher really didn't teach how the graphs would look like, I only know what y=(a/x) looks like. I don't really know how the numbers would affect the outcome of this graph.
Another way to find the range of - find its inverse (solve for x): xy= x- 4, x- xy= 4, x(1- y)= 4, .
If we think of that as a function of y, it is obvious that its domain is " . Since a function and its inverse "swap" domain and range, the range of the original function is "all y except 1".