graph the function below. Determine the domain, range, and horizontal asymptote. show work
f(x)=(7/5)^(x-3)
domain is all real numbers
range, as with all exponentials, is all reals > 0
horizontal asymptote is y=0
To graph the function f(x) = (7/5)^(x-3), we can start by picking some points and then connecting them to form a smooth curve. Let's choose a few values of x and find the corresponding y-values:
When x = 0, we have f(0) = (7/5)^(0-3) = (7/5)^(-3) = 5^3/7^3 = (125/343).
When x = 1, we have f(1) = (7/5)^(1-3) = (7/5)^(-2) = 5^2/7^2 = (25/49).
When x = 2, we have f(2) = (7/5)^(2-3) = (7/5)^(-1) = 5^1/7^1 = (5/7).
When x = 3, we have f(3) = (7/5)^(3-3) = (7/5)^0 = 1.
When x = 4, we have f(4) = (7/5)^(4-3) = (7/5)^1 = 5/7.
When x = 5, we have f(5) = (7/5)^(5-3) = (7/5)^2 = 25/49.
When x = 6, we have f(6) = (7/5)^(6-3) = (7/5)^3 = 125/343.
Now, we can plot these points on a graph:
(x, f(x)):
(0, 125/343)
(1, 25/49)
(2, 5/7)
(3, 1)
(4, 5/7)
(5, 25/49)
(6, 125/343)
By connecting these points, we get a smooth curve.
The domain of the function is all real numbers since the base (7/5) is positive for any real number value of x.
The range is the set of all positive real numbers because any positive exponent of (7/5) will result in a positive value.
The horizontal asymptote can be determined by observing that as x approaches positive infinity or negative infinity, the function approaches 0. Hence, the horizontal asymptote is y = 0.
To summarize:
- Domain: All real numbers
- Range: All positive real numbers
- Horizontal asymptote: y = 0