3.26: Completed square and logarithmic function g

0606/13/M/J/25 — Question 9 · 12 marks · Equations, Exponentials

The function

f

is defined by

f(x) = 2x^{2} + 9x - 10

for

0 \leqslant x \leqslant 3

(a)(i)

Write

f(x)

in the form

a + b(x + c)^{2}

where

a

b

and

c

are constants.

[3]

(a)(ii)

Hence determine whether or not

f^{-1}

exists.

[2]

(b)(i)

The function

g

is defined by

g(x) = 3\ln(5 - 2x)

for

0 \leqslant x < 2.5

On the axes, sketch the graph of

y = g(x)

. State the exact values of the intercepts with the coordinate axes and the equation of any asymptote.

[3]

(b)(ii)

Find an expression for

g^{-1}(x)

[2]

(b)(iii)

Find the domain and range of

g^{-1}

. Give each answer in exact form.

[2]