Question

Find the domain, range and asymptote of
y=10-e^(-x)

Answer 1

missing something ?

Answer 2

whatever it is suppose to be make sure what is inside the log is larger than zero
that is all

Answer 3

One sec. Wrong function

Answer 4

\[-x>0\] solve for \(x\)

Answer 5

So x is greater than 0?
Domain is x element of the real numbers?

Answer 6

@satellite73

Answer 7

domain is all the x values that you can plug into a function

Answer 8

look at the given expression, are there any x values that make the expression undefined ?

Answer 9

no?

Answer 10

so the domain is all real numbers : \((-\infty,~ +\infty)\)

Answer 11

Okay, how do you get the range?

Answer 12

find the limits as \(x\to+\infty\) and \(x\to -\infty\)

Answer 13

The range is the real numbers?

Answer 14

how ? what did u get for the limits ?

Answer 15

I wasn't really sure if I got the limits... but it looks like it goes infinity and neg. infinity..

Answer 16

"it looks like" is a forbidden phrase in calculus, work the limits and you will be surprised im sure

Answer 17

Oh... Can you please help with one of the limits? Just to make sure I'm doing it correctly..

Answer 18

\[\large{\begin{align}
\lim\limits_{x\to+\infty} 10-e^{-x} &= \lim\limits_{x\to+\infty} 10- \lim\limits_{x\to+\infty} e^{-x}\\~\\
&=\lim\limits_{x\to+\infty} 10- e^{\lim\limits_{x\to+\infty}-x}\\~\\
&=10- e^{-\infty}~~*\\~\\
&=10-0\\~\\
&=10\\~\\
\end{align}}\]

Answer 19

so the graph of given function flattens to 10 as you walk extreme right on x axis

Answer 20

work the other limit to find out what happens to the function as you go extreme left

Answer 21

how did e^-infinity turns out to be a zero?

Answer 22

good question, i have used exponent property :
\[\large a^{-n} = \frac{1}{a^n}\]

Answer 23

\[\large e^{-\infty} = \frac{1}{e^{\infty}} = \frac{1}{\text{think of very big positive quantity here}} = 0\]

Answer 24

oh I see.

Answer 25

It approaches to zero

Answer 26

So the other limit goes to infinity.

Answer 27

sure ?

Answer 28

does it go to "positive" infinity or "negative" infinity

Answer 29

negative infinity

Answer 30

good, can you guess the range now

keep in mind 10-e^(-x) is a continuous function

Answer 31

also it is a strictly increasing function

work the first derivative quick if you want

Answer 32

remember how to prove if a function is increasing ?

Answer 33

Not really...

Answer 34

But is the range this (-infinity,10)

Answer 35

In \([a,b]\), if \(f'(x)\) is positive, then \(f(x)\) is a strictly increasing in that interval.

Answer 36

yes range is (-infinity, 10)

Answer 37

So there's horizontal asymptote on y=10?

Answer 38

Yes, also thats clear from graph

Answer 39

I think I got it from here. Thank you for your help! :)

Answer 40

yw