fx bx and gx logbx are inverse functions Explain why the f

f(x) = b^x and g(x) = log_bx are inverse functions. Explain why the following is true.

The inverse of f1(x) = b^{(x – h)} is not equivalent to a vertical stretch or vertical compression of g

Solution

Given :f(x) = b^x and g(x) = log_b(x) are inverse functions

If f(x) and g(x) are inverse then ( f o g)(x) or (g o f)(x) =x

So, (go f )(x) = logb(b^x) = x*logb(b) (Plug x= b^x)

= x*logb/logb = x

So, the given functions are inverse functions.

f1(x) = b^{(x – h)}

y = b^(x-h)

Inverse of f1: x = b^(y-h)

Take log on both sides:

ln(x) = (y-h)lnb

ylnb = lnx + h*ln(b)

y = lnx/ln(b) + h

y = ln_x(b) +h

f^-1(x) = ln_x(b) +h

g(x) =  log_b(x)

We can see that f^-1(x) and g(x) are different functions and f^-1(x) cannotb be obtained by function transformations