Question

Graph y=5^x and y=log(sub)5x on a sheet of paper using the same sat of axes.
(i don't need the graph)
Use the graph to determine the domain and range of each function.
Then identify the y-intercept of each function and any asympotes of each function.

Answer 1

\[y=5^x\]

Answer 2

Were you able to determine the domain and range?

Answer 3

Nope, I don't know how to do the functions with graphs and all...

Answer 4

it turns out that because you can plug in any real number for x in y = 5^x, and you'll get some real number output for y, this means that the domain of y = 5^x is the set of all real numbers

Answer 5

basically, there are no restrictions for the domain of y = 5^x, which is another reason why the domain is the set of all real numbers

Answer 6

and because there are no domain restrictions, there are no vertical asymptotes

a vertical asymptote will only show up if you have a domain restriction of some kind

Answer 7

with me so far?

Answer 8

okay, so your saying since anything can be x, the domain would be all read numbers,
therefore there can't be an asymptote because you need some sort of restrictions in the domain for there to be one, right?

Answer 9

then what about the range?

Answer 10

exactly

the range is the set of all possible outputs, so because 5^x is ALWAYS positive, this makes the range to be the set of all positive real numbers

another way to put it: the range is the set of all numbers y such that y > 0

Answer 11

Summary:

Domain of y = 5^x is the set of all real numbers
Range of y = 5^x is the set of all real numbers y such that y > 0


In interval notation the domain and range are...

Domain: \(\Large (-\infty,\infty)\)
Range: \(\Large (0,\infty)\)

Answer 12

okay so the way to answer this would be by saying the domain can be all real numbers, and since there is no restriction on the domain, there cannot be any asypote, therefore the domain is all real numbers and the range is all positive real numbers or y>0 ?

Answer 13

correct,

domain is all real numbers ---> no vertical asymptote
range is y > 0 ----> horizontal asymptote is y = 0

Answer 14

to find the y-intercept of any function, you would plug in x = 0 and evaluate to find y

Answer 15

okay so if y=5^x and x=0 then y=1 right? which would mean the y-intercept is 1?

Answer 16

yes, which is the point (0,1)

Answer 17

thank you so much!

Answer 18

you're welcome