Question

Graph \(y=5^x~~~ and~~~ y=\log_{5}x\) on a sheet of paper using the same set of axes. Use the graph to describe the domain and range of each function. Then identify the y-intercept of each function and any asymptotes of each function.

Answer 1

@Data_LG2 :)

Answer 2

so i think we'll do \(\sf y=5^x\) first.
Looking at the graph, what happens to the x-values as it goes to negative infinity? positive infinity?

Answer 3

I'm not sure I understand....

Answer 4

it also goes from negative infinity to positive infinity right ? so therefore the domain is all set of real numbers

it means that whatever x-value you have to put in, there will be a corresponding y-value to it... for example if you plug a very large negative number, the y-value approaches 0 but never touches it (we'll discuss it later with the range) .. on the other hand if you plug in a very large positive number, you'll get a very large positive y value :)

Answer 5

any confusions in this part?

Answer 6

I think I got that part.....

Answer 7

okay good, so you'll say that the domain of \(\sf y=5^x\) is \(\ (x|x \in \mathbb{R} )\)...

now let's move on to the range or y-values of this function..
what happens to y-values as x approaches negative infinity? as x approaches positive infinity?

(oh i know why you get confused with the first one! my question should be what happens to the x-values as y approaches positive infinity and negative infinity, my bad sorry xD )

Answer 8

Don't they decrease?

Answer 9

quite right
as x approaches negative inifinity, y- approaches....? do y-values reach negative values?

Answer 10

No. Because once it hits 0, it doesn't decrease anymore. Right?

Answer 11

but does it really touch 0?

Answer 12

when x is -4.725, y reaches 0.

Answer 13

let's check it out:
\(\sf y=5^x = 5^{-4.725}\) plug it in the calculator and i got... 0.00049851575...

one of the properties of exponential functions is it never reaches 0 (unless you apply some vertical transformations)

so that's the first part of the range.. what happens when x approaches positive infinity? what happens to the y-values?

Answer 14

Oh! okay :) they increase

Answer 15

without limits or with limits?

Answer 16

without limits? I'm guessing.......I'm not completely sure because it just goes on and on....

Answer 17

yes absolutely! therefore the range is set of \( (y|y \in \mathbb{R}, y > 0 )\) because y can't be negative
so domain and range done..

y intercept, this means when x=0 what is the value of y? \(\sf y=5^x=5^0=?\)

Answer 18

It is 1.....

Answer 19

anything to the zero power is one :)

Answer 20

right (:

Answer 21

One quick question....is there a different way to write the range? I can't remember for the life of me :(

Answer 22

in words i guess? y is all set of real numbers where y must be greater than 0

in your class, how does your teacher write domain and range?

Answer 23

I'm not sure because I'm doing online school so the teacher never writes anything :P

Answer 24

heehee

Answer 25

oh i see.. that's how we write it here so yeah, im not sure whether there's another way :/

Answer 26

Okay :)

Answer 27

now for the next one, there isn't just one y intercept because it is x=0 for infinity.....

Answer 28

lol we're not done with the first one yet.. asymptote?

Answer 29

oh yeah :P whoopsies!

Answer 30

so what do you think is the asymptote of the function is?

Answer 31

y=0 or x-axis?

Answer 32

because no matter what power you do 5 by, y will never be 0....

Answer 33

yes exactly!! y=0 is the asymptote :D

Answer 34

Okie dokie artichokie! :D

Answer 35

Is that all for \(\Large y=5^x\)?

Answer 36

yes!

Answer 37

Okay...now for the weird(er) one :P

Answer 38

oh yeah, log -.-

okay we can do it by just looking at the graph and answering all the questions like before:
for the domain: what happens to the x-values as y approaches positive infinity and negative infinity ?
for the range: what happens to y-values as x approaches negative infinity? as x approaches positive infinity?

Answer 39

so the domain, the x values decrease until it reaches zero....right?

Answer 40

yes, as y values approach negative infinity, x values approaches zero
how about when y approaches positive infinity, what happens to x?

Answer 41

it increases

Answer 42

right?

Answer 43

wait...no.....

Answer 44

yes :) so the domain for this function is the range of the first function

Answer 45

oh!

Answer 46

which makes sense because these two functions are inverse of each other :)

Answer 47

and then the range is the domain of the first function?

Answer 48

yeppers

Answer 49

I'm not sure what to put for the y intercept because it would be (0, infinity) so I'm not sure

Answer 50

I know the asymptote will be x=0

Answer 51

Well there can't be a y-intercept because x can never be 0! :P

Answer 52

true :) you got it !

Answer 53

Okay thank you so much for helping me! I don't think I would have been able to figure it out myself :P

Answer 54

Okay just a quick check of my answer on the next one, would evaluating the logarithm of \[\log_{6}\frac{ 1 }{ 36 } \] just be \[\log_{6}\frac{ 1 }{ 36 } =-2\]

Answer 55

yes that's right, \(\sf 6^{-2}=\frac{1}{36}\)

Answer 56

okay great! Thanks again!!!

Answer 57

non problem :)

Answer 58

:)

Answer 59

I am going to work on Ariel when I'm done with my last math lesson before winter break :) just one more, but I think I got them :)

Answer 60

okay ^_^ goodluck!

Answer 61

gracias, mi amigo! :D

Answer 62

woohoo! done with math until after winter break! :D