Graph the function below. Determine the domain, range, and horizontal asymptote. Please show all of your work. Submit your graph to the Dropbox.
f(x) = (3/2)^(4-x)
To graph the function f(x) = (3/2)^(4-x), we can start by finding the domain and range of the function.
1. Determining the domain: The domain of a function is the set of all possible x-values for which the function is defined. Since there are no restrictions on the variable x, the domain of this function is all real numbers, or (-∞, ∞).
2. Determining the range: The range of a function is the set of all possible y-values. Since the base of the exponential function f(x) = (3/2)^(4-x) is positive, the function will always be positive. As x approaches infinity, the value of (4-x) approaches negative infinity, causing the exponential term to approach zero. As x approaches negative infinity, the value of (4-x) approaches positive infinity, causing the exponential term to approach infinity. Therefore, the range of this function is (0, ∞).
3. Determining the horizontal asymptote: The horizontal asymptote of an exponential function of the form f(x) = a^x, where a is a positive constant, is always y = 0. In this case, the base of the exponential function is (3/2), which is a positive constant. So, the horizontal asymptote for f(x) = (3/2)^(4-x) is y = 0.
Now, let's proceed to graph the function:
To summarize:
- Domain: (-∞, ∞)
- Range: (0, ∞)
- Horizontal asymptote: y = 0
To create the graph, you can use graphing software or an online graphing tool. Alternatively, you can manually plot points by choosing specific values for x and calculating the corresponding y-values. Choose a range of x-values that cover the domain of the function and plot the corresponding points on a graph.
Here is an example plot of the function f(x) = (3/2)^(4-x):
(Graphical representation not possible in text format)
Once you have created the graph using an appropriate tool, you can submit it to the Dropbox.