Question

The inverse function of f(x)=log(x) is f

−1
(x)=
log(x)
1

. (b) If a and b are positive constants, then y=ln(ax+b) has no vertical asymptote.

Answer 1

The inverse of the logarithmic function f(x) = log(x) is the exponential function f^-1(x) = e^x. The natural logarithm and exponential functions are inverses and can be used to simplify expressions. The function y = ln(ax + b), with 'a' and 'b' as positive constants, lacks a vertical asymptote.

Step-by-step explanation:

The question revolves around the concept of inverse functions, specifically focusing on the relationship between exponential functions and their inverses, the natural logarithms (ln). When dealing with the logarithmic function f(x) = log(x), its inverse is the exponential function f-1(x) = ex. A mathematical 'trick' often used involves recognizing that natural logs and exponentials cancel each other out, so that ln(ex) = x and eln(x) = x. This property of logarithms is essential when dealing with growth calculations and transformations.

The function y = ln(ax + b), where a and b are positive constants, does not have a vertical asymptote as long as the argument (ax + b) remains positive, because the natural log function only takes positive arguments, and a linear function with a positive slope will never be zero or negative if 'b' is positive.