What is the inverse of the function y= e the power of x when 0 less than or equal to 1 and their graph
To find the inverse of a function, we need to first solve for x in terms of y. For the function y = e^x, taking the natural logarithm of both sides gives:
ln(y) = x
So the inverse function is:
x = ln(y)
Now, we can graph both the original function and its inverse on the same set of axes to see how they relate:
As we can see from the graph, the original function y = e^x is an increasing exponential function that passes through the point (0,1). Its inverse, x = ln(y), is a decreasing logarithmic function that passes through the same point. The two functions are reflections of each other across the line y = x, which represents the identity function.
To find the inverse of a function, we need to switch the roles of x and y and solve for the new y.
Given the function y = e^x, we want to find its inverse.
Step 1: Switching the variables
x = e^y
Step 2: Solve for y
Taking the natural logarithm (ln) of both sides:
ln(x) = ln(e^y)
Using the property of logarithms, ln(e^y) simplifies to y:
ln(x) = y
Therefore, the inverse of the function y = e^x is y = ln(x).
Now, let's take a look at their graphs.
The graph of y = e^x is an exponential curve that passes through the point (0, 1) and increases rapidly as x increases.
The graph of y = ln(x) (also known as the natural logarithm) is a curve that passes through the point (1, 0) and increases slowly as x increases.
These two graphs are reflections of each other across the line y = x since they are inverse functions of each other.