Interesting question I'm confused about "Which 2 functions grow and different rates? A. x^2+4x and x^2 B. ln(x) and ln(x^2) C. e^(x+100) and e^(x-100) D. x^3 and x^4 t…

Question

Interesting question I'm confused about

"Which 2 functions grow and different rates?

A.
`x^2+4x` and
`x^2`

B.
`ln(x)` and
`ln(x^2)`

C.
`e^(x+100)` and
`e^(x-100)`

D.
`x^3` and
`x^4`

the answer is D, but WHY is that the answer and how do you get it
explanation required

Answer 1

This is probably a question about comparing the specific operations being performed in the pairs of functions provided. In all cases but D, the rate of growth converges at extremes of x. Tbh it could probably be worded better...

In A, the degree of both expressions is the same: 2. The biggest thing being done to x that it's squared, and the fact that one expression has an extra x term doesn't change it's rate of growth significantly because at high values of x, the functions will essentially be the same (e.g. compare 1000^2+4(1000) and 1000^2 - the difference is quite small and only gets smaller as you go up)

In B, again the same operation is being performed (taking the natural logarithm of x). The second expression can also be written as 2lnx, making it quite similar to the first (but with a vertical stretch by a factor of 2).

Same idea again for C - same operation (e^x) and same order of operation being performed.

For D, x^3 and x^4 are different functions completely. As you increase x, the functions begin to diverge from one another more and more. This reflects significantly different rates of growth.