what does it mean when it says describe the end behavior of f and whats that for f(x)=(x^2+x+3)/(x-1)

and f(x)= 3*2^x,
please explain everything, and if you use --> tell me what that means, cuz i have no clue

--> means "goes to"

as in what happens to your function there as x goes to 1
or as x --> 1

As x -->1
The numerator becomes 1+1+3 = 5
so the function looks like 5/(x-1)
but of course x-1 is 0 when x =1 so the function becomes undefined when x = 1
but what if x = 1.001 for example
then the function is about
5/.001 or 5000
and if x = .0001
then it is about 50,000 very big
Now look at x = .999
then it is 5/-.001 = -5000
and if x = .9999 then f = -50,000
do we see a trend here?
graph it.
Now what happens as x-->oo
when x is big, x^2 is much bigger than x or 3
so the numerator looks like x^2
similarly the denominator looks like x
so for big +x the function looks like just plain old x, a straight line of slope +1
And if you do x-->-oo
then the numerator is positive x^2 and the denominator looks like x so the result is again looks like x, that same straight line of slope +1

what is the big +x?

and so the answer is
"so for big +x the function looks like just plain old x, a straight line of slope +1" and "then the numerator is positive x^2 and the denominator looks like x so the result is again looks like x, that same straight line of slope +1"? if so is there an easier way to write that?

what is the big +x?

In other words what happens when x is a large positive number.

ok, that's wahat i thought but wasn't sure.

When we are asked to describe the end behavior of a function, we are essentially examining what happens to the function's output as the input approaches positive or negative infinity. In other words, we want to understand the long-term trends of the function as the input values get extremely large or extremely small.

Let's break down the end behavior of the given functions:

1. f(x) = (x^2 + x + 3)/(x - 1)

To analyze the end behavior of this rational function, we can look at the degrees of the numerator and denominator. In this case, the degree of the numerator is 2, represented by the term x^2, and the degree of the denominator is 1, represented by the term x.

When the degree of the numerator is greater than the degree of the denominator, like in this case, the end behavior is determined by the highest degree term in the numerator. In f(x), the highest degree term is x^2.

As x approaches positive or negative infinity, the term x^2 dominates the function, and the other terms become relatively insignificant. Therefore, the end behavior of f(x) can be described as:

As x approaches positive infinity (x --> +∞), the function approaches positive infinity. This can be written as lim f(x) = +∞ (as x --> +∞).

As x approaches negative infinity (x --> -∞), the function also approaches positive infinity. This can be written as lim f(x) = +∞ (as x --> -∞).

2. f(x) = 3 * 2^x

Now let's analyze the end behavior of this exponential function. The base of the exponent in f(x) is 2, and there is a constant factor of 3 multiplied by it.

For exponential functions with a positive base greater than 1, the end behavior is as follows:

As x approaches positive infinity (x --> +∞), the function approaches positive infinity. This can be written as lim f(x) = +∞ (as x --> +∞).

As x approaches negative infinity (x --> -∞), the function approaches zero. This can be written as lim f(x) = 0 (as x --> -∞).

Therefore, the end behavior of f(x) = 3 * 2^x is that the function grows rapidly towards positive infinity as x increases, and approaches zero as x decreases without bound.