determine the domain, range and horizontal asymptote Please show all your work
f(x) = - (3/4)^x - 1
Help!
domain : any real value for x
range : y > -1
horizontal asymtote: y = -1
domain and range, just look at a sketch of the graph
last part:
intuitively, as x --> ∞+ , value of (3/4)^x --> 0
( e.g. (3/4)^100 = 3.2 x 10^-13 , pretty close to zero)
as x -- > -∞
we get (3/4)^-x
= (4/3)^ +x ---> ∞
so look at x --> +∞
f(x) --> 0 - 1 = -1
so y = -1 is the horizontal asymtote
To determine the domain, range, and horizontal asymptote of the given function f(x) = - (3/4)^x - 1, we need to follow a few steps.
1. Domain:
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In this case, since we have an exponential function with a base of 3/4 raised to the power of x, the domain is all real numbers (-∞, ∞). There are no restrictions on the input values.
2. Range:
The range of a function represents the set of all possible output values (y-values) that the function can produce. In this case, as x approaches positive infinity, the value of (3/4)^x approaches 0. As x approaches negative infinity, the value of (3/4)^x approaches infinity. Thus, the range of this function is (-∞, -1].
3. Horizontal Asymptote:
To determine the horizontal asymptote, we need to examine what happens to the function as x approaches positive infinity and negative infinity. Since the exponent of (3/4)^x is negative, as x approaches positive infinity, the function approaches 0. Therefore, the horizontal asymptote of this function is y = 0.
In summary:
- Domain: (-∞, ∞)
- Range: (-∞, -1]
- Horizontal Asymptote: y = 0