Simplify the function. than determine the key aspects of the function

To simplify the function, we need to know some basic identities involving exponents.

1. b^(ax)=(b^x)^a=(b^a)^x

2. b^(x/d) = (b^x)^(1/d) = ((b^(1/d)^x)

Now simplify f(x), where

f(x)=(1/3)*(81)^(3*x/4)

=(1/3)(3^4)^(3*x/4) [ 81=3^4 ]

=(1/3)(3^(4*3*x/4) [ rule 1 above ]

=(1/3) (3^(3*x)

=(1/3)(3^(3x)) [ or (1/3)(27^x), by rule 1 ]

(A) Initial value is the value of the function when x=0, i.e.

initial value

= f(0)

=(1/3)(3^(3x))

=(1/3)(3^(3*0))

=(1/3)(3^0)

=(1/3)(1)

=1/3

(B) the simplified base base is 3 (or 27 if the other form is used)

(C) The domain for an exponential function is all real values ( - ∞ , + ∞ ).

(D) The range of an exponential function with a positive coefficient and without vertical shift is ( 0, + ∞ ).

the furthest the dog can be is 508 meters away from the house, and the closest he can be is 492 meters away from the house.