Determine the domain, range and horizontal asymptote

Question

Sorry, for the mistake in writing the question. I have never used this site before and more confused in this type of math. Thanks for any help. 

 

what is the domain and range and horizontal asymptote of

 

 f(x) = 5^-x -4

Answer 1

The domain of the function is all real numbers because the domain of the exponential function is all real numbers.   The range can be determined by investigating the end behavior.    As x approaches - infinity, the function itself approaches + infinity.   As x approaches + infinity, the function itself approaches -4, but does not quite get there.     Thus the range is   all real numbers y  such that y > -4.

 

The end behavior as x approaches + infinity (approaching -4 but never getting there) also shows us that there is a horizontal asymptote:   namely the straight line    y = -4.

 

Answer 2

The horizontal asymptote tells, roughly, where the graph will go when x is really, really big. So I'll look at some very big values for x in both the positive and negative direction

 

lets use x at ±1  ±10  ±100 and  ±10,000 and ±1,000,000

lets begin with the largest magnitude

 

x                    f(x)

1,000,000       5^-1000000 - 4  = 0 -4 ⇒ -4

10,000           5^-10000 - 4 = 0 - 4 ⇒ -4

100                5^-100 - 4 = 1.3x10^-70 - 4 ⇒ -4

10                  5^-10 - 4 = 0.0000001024 - 4 = -3.9999998976

1                    5^-1 - 4 =  0.2 - 4 = -3.8

0                    5⁰ - 4 = 1 - 4 = -3

-1                   5^-(-1) - 4 = 5¹ - 4 = 5 - 4 = 1    (remember the neg of a neg number is the pos)

-10                 5¹⁰ - 4 = 9765625 - 4 = 9765621

-100               5¹⁰⁰ - 4 =7.89x10⁶⁹ - 4 ⇒ 7.89x10⁶⁹

-10,000           5¹⁰⁰⁰⁰ - 4 = positive infinity - 4 ⇒ positive infinity

-1,000,000      5^1000000 - 4 = positive infinity -4 ⇒ positive infinity

 

(this weblink will give you a plot of the f(x)  http://fooplot.com/plot/8vjsus36fa)

 

Well when x is really large and positive your function is close to -4 and gets closer and closer to -4 as x increases.

 

However, when x is less than zero f(x) increases very fast to larger and larger values

 

So the horizontal asymptote would be -4 as it gets close but never really reaches -4

 

Now for your your domain that would be all values of x that f(x) is a valid function

 

there is no value of x that wouldn't get a vaue for f(x) so your domain would be

-infinity < x < infinity (all real numbers)

 

Now the range would be what values of f(x) are attainable. if you look at the graph, you know that

• for values of x >0 f(x) goes towards -4 but never reaches it
• for values of x<0 f(x) increases toward infinity

So your range would be all real numbers >-4

 

hope this helps

 

you can look at purple math for help with the definitions

http://www.purplemath.com/modules/asymtote.htm

http://www.purplemath.com/modules/fcns.htm

 

and this website to graph your functions to give you a good visual

fooplot.com

 

Good luck

PS no issue with the writing of your equation