HELPP
the equation is y=(1/3)^x

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.