The following continuous situations involve functions and conic functions. For e

Question

The following continuous situations involve functions and conic functions. For each assume that you pick real numbers a and b at random between o and 10. Find the probability of the following.

1. The function y = abx models growth

2. The Function Y = (1/(x-a)) + b has a horizontal asymptote

Explanation / Answer

When a > 0 and b is between 0 and 1, the function y = abx will be a decaying function. When a > 0 and the b is greater than 1, the function y = abx will be a growth function. Here, since both a and b are real numbers between 0 and 10, apparently a > 0.Since the real numbers between 1 and 10 are infinite, the probability of y = abx being a growth function can be computed only if b is an integer between 0 and 10. If b is an integer, the probability of b being greater than 1 is 8/9 (both 0 and 10 are excluded as both a and b are between 0 and 10). Thus, the probability of y = abx being a growth function is 8/9 or, 0.89 (on rounding off to 2 decimal places). The function y = [1/(x –a)] + b or y = [1 + b(x–a)]/(x-a) = (bx + 1 -ab )/(x-a)is a rational function. Since both the numerator and the denominator are of degree 1 and since the coefficients of x in the numerator and the denominator are band 1 respectively, hence the function has a horizontal asymptote which is the line y = b/1 or y = b.

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