Determine the domain,range and horizontal asymptote of f(x)=(3/2)^(4-x). Graph the function
typical exponential function
domain: any real value of x
range : y >0
as x---> +∞ , f(x) ----> 0
so the x-axis is a horizontal asymptote for large x
e.g. (3/2)^-500 = 9.0 x 10^-89 , awfully close to zero
as x ---> - ∞ , f(x) ---> ∞
e.g. (3/2)^500 = 1.11 x 10^88 , pretty big
To determine the domain, range, and horizontal asymptote of the function f(x) = (3/2)^(4-x), let's break down each of these components.
1. Domain:
The domain of a function refers to all the possible values of x for which the function is defined. In this case, the function f(x) = (3/2)^(4-x) is a power function with a base of (3/2). For a power function, the domain is typically all real numbers, except when the base is negative or zero. However, since the base in our function is positive (3/2 is positive), the domain is all real numbers. So, the domain of this function is (-∞, ∞).
2. Range:
The range of a function refers to all the possible values that the function can output (the y-values). In this case, since the base of the function f(x) = (3/2)^(4-x) is positive (3/2 is positive), the range will be all positive real numbers greater than zero. So, the range of this function is (0, ∞).
3. Horizontal Asymptote:
To find the horizontal asymptote of a function, we examine the behavior of the function as x approaches positive or negative infinity. In this case, as x approaches positive infinity (x → ∞), the exponent (4-x) approaches negative infinity, which makes the value of the function f(x) tend to zero. Therefore, the horizontal asymptote of this function is y = 0.
Now, let's graph the function f(x) = (3/2)^(4-x):
Since this is an exponential function with a base greater than 1 (3/2 > 1), the graph will start at the y-intercept (when x = 0) and increase as x moves to the left. As x moves toward negative infinity, the function will approach the x-axis (horizontal asymptote).
Please note that I am an AI text-based bot and do not have the capability to generate visual graphs. It's recommended to use graphing software or a graphing calculator to plot the graph for a better visualization.