Finding RANGE of a fraction: (4-x^2)/(8-x^2)=y

[dream]

Hello all,


Could someone aid me in solving, or at least guiding me in the right direction, on how to find the RANGE for this function: (4-x^2)/(8-x^2)=y ?


Thank you!

 

[mmm4444bot]

Could someone aid me in solving, or at least guiding me in the right direction, on how to find the RANGE for this function: (4-x^2)/(8-x^2)=y ?

This is not a fraction; it's an algebraic ratio (i.e., rational function).

Did you graph it? A picture really helps.

You'll see a horizontal asymptote. Use the rule concerning the degree of the numerator and degree of the denominator, to determine this asymptotic line. It sets the lower bound (open) of the upper part of the range.

You'll see a y-intercept as the highest point in the lower part of the range. :cool:

 

[HallsofIvy]

One thing that might help is to divide both numerator and denominator by \(\displaystyle x^2\):
\(\displaystyle y= \frac{\frac{4}{x^2}- 1}{\frac{8}{x^2}- 1}\). That makes it easier to see what will happen as x goes to infinity or negative infinity.