Does squaring a function affect the asymptotes?
For example, if you square the function 1 over x + 3 does it change the asymptotes that were y = 0 and x = -3?
Thank you!
This kind of question is best tackled with explorative tools such as Desmos.
Go to
https://www.desmos.com/calculator
then type in
1/(x+3) [enter]
1/(x+3)^2 [enter]
You will see the effect of squaring (blue curve) on the original (red curve).
Experiment with other functions and enjoy. An image is worth a thousand words.
Squaring a function does not change the asymptotes of the original function. In the given example, the original function is 1 / (x + 3), which has asymptotes at y = 0 and x = -3.
To understand why squaring a function does not affect the asymptotes, we need to consider the definition of asymptotes. Asymptotes are lines that a function approaches but never crosses as x tends to positive or negative infinity.
When we square a function, such as squaring 1 / (x + 3), we obtain (1 / (x + 3))^2 = 1 / (x + 3)^2. Squaring the function only affects the values of the function itself, not the behavior as x approaches infinity or negative infinity.
Therefore, the asymptotes remain the same: y = 0 and x = -3.