Question

Write down a transcendental function that has a y-intercept at (0,2), is always increasing and has a horizontal asymtote at y=0.

Answer 1

A transcendental function with a y-intercept at (0,2), that is always increasing, and a horizontal asymptote at y=0 can be represented by f(x) = 2e^-x. This function passes through the point (0,2), increases as x increases, and approaches zero as x approaches infinity.

Step-by-step explanation:

To write down a transcendental function that has a y-intercept at (0,2), is always increasing, and has a horizontal asymptote at y=0, we can consider an exponential decay function multiplied by a vertical shift factor. A suitable function meeting these criteria can be expressed as:

f(x) = 2e-x

This function has the following properties:

• It passes through the point (0,2) because f(0) = 2e0 = 2.
• It is always increasing because the exponential function e-x is decreasing, but since it has a negative exponent, it makes the overall function increase as x increases.
• It has a horizontal asymptote at y=0 because as x approaches infinity, e-x approaches zero, thereby causing the function f(x) to approach zero.