Exponential functions Flashcards
How is an exponential function defined?
f(x) = b^x
where b > 0 and b /= 1
Describe some characteristics of exponential functions with b > 1 when graphed
The domain of each function consists of all real numbers, and the range consists of all positive real numbers. Each graph has y-intercept (0,1). Moreover, the graphs have the same general shape. Each rises from left to right. As x increases, f.x/ also increases. In fact, f.x/ increases without bound. Equations with greater bases will show a quicker rise. Looking at quadrant II, we see that as x becomes very negative, the graphs of both functions approach the x-axis. We say that the x-axis is an asymptote for each graph. This implies that the function values get very close to 0.
How does the grap and function change when 0 < b < 1?
Still, the domain consists of all real numbers, and the range consists of all positive real numbers. The graph has y-intercept (0,1). However the graph here falls from left to right. That is, as x increases, f .x/ decreases. Notice that as x becomes very positive, f.x/ takes on values close to 0 and the graph approaches the x-axis. However, as x becomes very negative, the function values are unbounded.
Are exponential functions one to one?
The graph of an exponential function has one of two shapes, depending on the value of the base, b. It is important to observe that in either case the graph passes the horizontal line test. Thus, all exponential functions are one-to-one.
How would you plot (2^x) - 3?
The function has the form f(x) - c, where f(x) = 2^x and c = 3. Thus, its graph is obtained by shifting the graph of f.x/ D 2x three units downward.
