use transformation to graph the following function - state domain, range and horizontal asymptote f(x)= e^5x.
To graph the function f(x) = e^5x and determine its domain, range, and horizontal asymptote, we need to understand the properties of the exponential function.
1. Domain: The domain of an exponential function is all real numbers, so there are no restrictions on x for f(x) = e^5x.
2. Range: The range of an exponential function is all positive real numbers. In this case, since the base e is a positive number (approximately 2.718), raising it to any power will still give a positive value. Hence, the range of f(x) = e^5x is (0, ∞).
3. Horizontal Asymptote: To find the horizontal asymptote, we need to examine the behavior of the function as x approaches positive infinity or negative infinity. In this case, as x approaches negative infinity, e^5x approaches zero since the exponent becomes very large and negative, making the value of the function very small. Therefore, the horizontal asymptote is y = 0.
Now, let's graph the function using transformations:
1. Start with the parent function y = e^x, which has a horizontal asymptote at y = 0.
2. Apply the transformation of stretching the graph horizontally by a factor of 5. This means that for each unit increase in x, the y-value will increase at a much faster rate than the parent function.
3. The graph of f(x) = e^5x will increase more steeply than the parent function as x increases. It will never touch or cross the x-axis and will approach the horizontal asymptote at y = 0 as x approaches negative infinity.
The resulting graph will resemble an upward-sloping curve that increases more rapidly compared to the base exponential function y = e^x.
Remember to label the x-axis, y-axis, and mark important points, such as intercepts or points of interest for a complete graph.