f: x |→ ex + 2 , x ∈ ℜ
g : x |→ ln x , x > 0
(a) State the range of f.
(b) Find fg(x) , giving your answer in its simplest form.
(c) Find the exact value of x for which f(2x +3) = 6
(d) Find f−1 , the inverse function of f, stating its domain.
(e) On the same axes sketch the curves with equation y = f(x) and y = f-1(x) coordinates of all the points where the curves cross the axes.
(ii) (1 - 10x)/(2x - 1)5 giving your answer in its simplest form.
(b) Given that y = 3 tan 2y, find dy/dx, in terms of x.7 (a)(i) Product Rule/Chain Rule
Given that f(x) = R cos(2x + α) , where R > 0 and 0 < α < 90°,
(a) find the value of R and the value of α.
(b) Hence solve the equation
f(x) = 7 cos 2x - 24 sin 2x = 12.5
for 0 ≤ x < 180° , giving your answers to 1 decimal place.
(c) Express 14 cos2x - 14 sinx cosx in the form a cos 2x + b sin 2x + c, where a, b, and c are constants to be found.
(d) Hence, using your answers to parts (a) and (c), deduce the maximum value of
14 cos2x - 14 sinx cos
8 (a) Rcos( ) method
The curve passes through the points P(−1.5, 0) and Q(0, 5P) as shown.
On separate diagrams, sketch the curve with equation
(a) y = |f(x)|
(b) y = f(|x|)
(c) y = 2f(3x)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
(b) Hence show that
4cosec22θ cosec2θ = sec2θ
(c) Hence or otherwise solve, for 0 < θ < π,
4cosec22θ cosec2θ = 4
giving your answers in terms of π.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.