Create a function which has the following properties:
a. It has a horizontal asymptote at y=2
b. It has a discontinuity at x=2 which is not a vertical asymptote
c. It has no other discontinuities or asymptotes
if x>/=0, y = 2-2e^-x
if x <2, y = -1
Could you explain, in detail, why each of the properties are satisfied by your example?
To create a function that satisfies the given properties, we can follow these steps:
Step 1: Start with a basic function. Let's use the function f(x) = 1/x as a starting point.
Step 2: Shift the function vertically to have a horizontal asymptote at y=2. To do this, we can add 2 to the original function: f(x) = 1/x + 2.
Step 3: Create a discontinuity at x=2. Since we want a discontinuity that is not a vertical asymptote, we can use a removable discontinuity or a hole. One way to create a hole is by canceling out a removable factor. In this case, we can cancel out the factor (x - 2) in both the numerator and denominator, resulting in: f(x) = (x - 2)(1/x + 2)/(x - 2) = 1 + 2(x - 2)/x.
Step 4: Simplify the function if possible. We can simplify further by expanding the numerator: f(x) = 1 + 2x/x - 4/x = 1 + 2 - 4/x = 3 - 4/x.
The final function that meets all the given properties is: f(x) = 3 - 4/x.