Consider the function

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

f(x)=\ln(3x+5)

, which has domain

x>-\dfrac{5}{3}

. (a) Verify that

y=0

lies in the range of

f

and determine the corresponding

x

-value

a

for which

f(a)=0

. (b) Calculate

(f^{-1})'(0)

. Show each algebraic step and provide the exact value. (c) Using your result from part (b), write the equation of the line tangent to the graph of

y=f^{-1}(x)

at the point where

x=0

. Simplify completely.