For a function y = f(x), the domain is the set of the possible values of x that we can input, while the range is the set of the possible outputs of the function.
In this case we gave y = f(x) = (1/3)^x
A) Domain
To see the domain we always start assuming that the domain is the set of all real numbers, and then see if there is a real number that causes any problem.
(for example if we had a denominator like "x - 2", the point x = 2 would cause problems because it would make the denominator equal to zero, and we can not divide by zero)
Particularly because x is the exponent, we do not have any problem here, so the domain is the set of all real numbers:
D: = R
B) Range.
To find the range we usually need to find the smallest value of the function.
Here we have:
y = (1/3)^x
Is really easy to see that as x increases, the denominator increases but the numerator remains the same, then as x increases, the value of y decreases.
For example for x = 100 we will have:
y = 1/3^100 ≈ 0
(you also can see that y will never be a negative value, and as x increases the value of y tends to zero)
then:
0 < y
And if we have x negative, like x = -3 for example.
y = (1/3)^-3 = (3/1)^3 = 3^3 = 27
So if x decreases (or increases in the negative value) then the value of y increases, so we do not have any upper limit.
Then the range is only defined by
0 < y.
We can write the rane as:
R: (0, ∞)
C) The y-intercept is the value of y when x= 0
y = (1/3)^0 = 1
(for any number different than zero N, we have that N^0 = 1)
the y-intercept is 1
D) The asymptote is a tendency to a given value, such that the function never does reach that value.
Here we have two asymptotes.
y tends to infinity as x tends to negative infinity.
y tends to zero as x tends to infinity.
