Mathematics

Problem

A curve representing

f(x)=k b^{x}

passes through the points

(0,9)

and

\bigl(2,\tfrac{9}{4}\bigr)

. a. (i) Determine the constants

k

and

b

. b. (ii) Prove that

f(x)

is decreasing for all real

x

. c. (iii) Find the value of

x

for which

f(x)=\tfrac{9}{16}

. d. (iv) Sketch the graph, clearly indicating the

y

-intercept, the value obtained in part (iii) and the horizontal asymptote.