graph the function below. Determine the domain, range and horizontal asymptote. f(x) = -(3/4)^x - 1
as with all exponentials, the domain is all real numbers.
All exponential curves look basically alike. If n > 1, n^x curves up to the right.
If n < 1 the graph curves down to the right.
n^x always has the x-axis as an asymptote.
Note that n must be positive, since n^x = e^(x ln n) and ln is not defined for n<0.
So, the above graph looks like e^-x, but is flipped upside down because of the negative sign.
Then the graph is shifted down one unit, so it has y=-1 as the asymptote.
It rises steeply from the left, goes through (0,-1) and approaches the line y = -1 from below.
sorry - goes through (0,-2)
It's ok figured that out on my own :)
To graph the function f(x) = -(3/4)^x - 1, follow these steps:
Step 1: Determine the domain.
The domain of a function represents all the possible x-values for which the function is defined. In this case, there are no specific restrictions, so the domain of f(x) is all real numbers.
Step 2: Determine the range.
The range of a function represents all the possible y-values that the function can take. To determine the range, we need to analyze the behavior of the function.
Note that the base of the exponential term (3/4) is between 0 and 1, which means as x approaches positive infinity (x → +∞), the value of f(x) will approach 0. Additionally, as x approaches negative infinity (x → -∞), the value of f(x) will become a large negative number. So, the range of f(x) is (-∞, 0).
Step 3: Determine the horizontal asymptote.
The horizontal asymptote of a function represents the horizontal line that the graph approaches as x tends to positive or negative infinity. For exponential functions of the form f(x) = a^x, where -1 < a < 1, the horizontal asymptote is always y = 0.
In this case, since the base of the exponential term is (3/4), which satisfies the conditions -1 < a < 1, the horizontal asymptote of f(x) is y = 0.
Step 4: Graph the function.
To graph the function, you can select several x-values, calculate the corresponding y-values, and plot the points on a graph. Alternatively, you can use a graphing calculator or software to obtain an accurate graph.
Remember, as x approaches positive infinity, the function will approach the horizontal asymptote at y = 0. As x approaches negative infinity, the function will become a large negative number.
Using these steps, you should be able to graph the function and identify its domain, range, and horizontal asymptote.