Question

Is y = 0 the asymptote of all functions of the form f(x) = ab^x? Explain your reasoning.

Answer 1

No, y = 0 is not the asymptote of all functions of the form f(x) = ab^x. The y = 0 line is called the x-axis and is only an asymptote for certain functions. Functions of the form f(x) = ab^x do not have y = 0 as an asymptote.

Step-by-step explanation:

No, y = 0 is not the asymptote of all functions of the form f(x) = ab^x. The y = 0 line is called the x-axis and is only an asymptote for certain functions. In order for a function to have y = 0 as an asymptote, the limit of the function as x approaches infinity or negative infinity would need to be 0. For example, the function f(x) = 1/x has y = 0 as an asymptote because as x approaches infinity or negative infinity, the function approaches 0. However, functions of the form f(x) = ab^x do not have y = 0 as an asymptote because as x approaches infinity or negative infinity, the value of the function either approaches infinity or approaches 0, depending on the values of a and b.

Answer 2

y=0 would be a horizontal line. An asymptote is a line that a function approaches, but never reaches. Exponential functions such as these are a smooth curve. If both numbers are positive numbers greater than or equal to 1, the curve increases. If at least one of the numbers is a positive number between 0 and 1, the curve decreases. If a is a negative number, the curve decreases as well. If either a or b is zero, then the graph would stay constant at 0. However, as long as neither a nor b is zero, then this graph will never touch that point. The only way to get an answer of y=0 is to multiply by 0. If neither a nor b is zero, this won't happen.