Question

find the end behavior of f(x)=e^(x-3)-3

Answer 1

I think you have to take the natural log :)

Answer 2

no

Answer 3

At first, you have a shift ( from \(\large\color{slate}{ f(x)=e^{x} }\) )

3 units right \(\large\color{slate}{ f(x)=e^{x\color{blue}{-3}} }\)
3 units down, \(\large\color{slate}{ f(x)=e^{x-3} \color{blue}{-3} }\)

lets, just observe the \(\large\color{slate}{ f(x)=e^x }\) for now, though.

Answer 4

1)
When x gets infinitely small, (\(\large\color{slate}{ -10 }\) , \(\large\color{slate}{ -100 }\) , \(\large\color{slate}{ -1000 }\) .... \(\large\color{slate}{ -\infty }\))
THEN, what happens to the y?


2)
When x gets infinitely large, (\(\large\color{slate}{ 10 }\) , \(\large\color{slate}{ 100 }\) , \(\large\color{slate}{ 1000 }\) .... \(\large\color{slate}{ +\infty }\))
THEN, what happens to the y?

Answer 5

\(\large\color{blue}{ f(x)=e^{x} }\)

Can you evaluate the \(\large\color{blue}{ f(x) }\) when x=-10 ?

Answer 6

(you can just tell me the exact value)

Answer 7

1/e^10?

Answer 8

yes.

Answer 9

So, for x=-100 and x=-1000, it would be,

\(\large\color{blue}{ f(-100)=1/e^{100} }\)
\(\large\color{blue}{ f(-1000)=1/e^{1000} }\)

Answer 10

Which has a smaller value, \(\large\color{blue}{ 1/e^{100} }\) or \(\large\color{blue}{ 1/e^{1000} }\) ?

Answer 11

1/e^100

Answer 12

Which is smaller, 1/100 or 1/1000 ?

Answer 13

1/1000

Answer 14

that made a little more sense

Answer 15

Wouldn't \(\large\color{blue}{ 1/{1000} }\) be smaller than \(\large\color{blue}{ 1/{100} }\) ?

Answer 16

yes, and so,
\(\large\color{blue}{ 1/e^{1000} }\) is smaller than \(\large\color{blue}{ 1/e^{100} }\)

Answer 17

So this is what we have: \(\large\color{blue}{ f(x)=e^{x} }\)

\(\large\color{blue}{ f(-10)=1/e^{10} }\)
\(\large\color{blue}{ f(-100)=1/e^{100} }\)
\(\large\color{blue}{ f(-1000)=1/e^{1000} }\)
\(\large\color{blue}{ f(-10000)=1/e^{10000} }\)

Answer 18

yeahh

Answer 19

So, as x becomes a very big negative, like -100, -1000, -10000 and on,
the denominator is increasing, and thus the f(x) is approaching zero closer and closer

Answer 20

\(\large\color{blue}{ 1/e^{100} }\) is pretty much zero.
But \(\large\color{blue}{ 1/e^{10000} }\) is much closer to zero.
(it has even less value)

Answer 21

So, as x approaches \(\large\color{blue}{ -\infty}\), the f(x) approaches \(\large\color{blue}{ 0 }\)

Answer 22

is this making sense?

Answer 23

(\(\large\color{blue}{ -\infty }\) means negative infinity)

Answer 24

Now, as x becomes a very big number, where is the y value going?

Answer 25

yes

Answer 26

\(\large\color{blue}{ f(x)=e^x }\)

as x is, 10, 100, 1000 and on, is y increasing or decreasing?

Answer 27

sorry, but I am asking if f(x) becomes very large or very small, as x values are growing?

Answer 28

(what do you think ?)

Answer 29

decreasing