The ocean pressure, P, under sea level can be modelled by wd pk xv;e4-/othe function
$P(D)=\frac{D}{10}​+1$
where D is the distance in metres below sea level and P is measured in atmospheres.
A submarine cruising near the surface is submerged according to the function
$D(t)=10+5t$
where t is measured in minutes and D is the distance the submarine is below sea level, measured in metres.
1.Find the composite function $P∘D$ and explain what it means in the
context of this question
2.Find and interpret $(P∘D)(10)$.

The temperature, T, measured in degrees Celsius, above ground level in y )aaov4*q7yc Bagan, Myanmar, can be modelled by the yv a) qycoa4*7function
T(H)=37−$\frac{H}{160}$
where H is the height above ground, measured in metres.
The height, H, of a hot air balloon carrying tourists on a particular day in Bagan is given by the function
H(t)=40+10
twhere H is in metres, and t is the time in minutes after reaching cruising height.
1.Find the composite function T∘H and explain what it means in the
context of this question.
2.Find $(T∘H)(36)$ and explain what it means in the context of this
question.

A tyre manufacturing company has found that the ae+cd* sa0 dx( i2roy;number of tyres it produces, N, can be modelled bys2 0+c(diodxae r *a;y the function
N(t)=3t−9where t is the number of hours the factory operates per day, with a minimum of 3 hours.
The profit the company makes, P, in dollars, depends on the number of tyres produced, and is modelled by the function
(N)=60N−850where N is the number of tyres produced.
1.Find the company's profit or loss if it operates for 66 hours per day.
2.Find the company's profit in terms of the hours of operation per day, t.
3.Determine the number of hours the company needs to operate the factory per day in order to earn a positive profit. Give your answer to the nearest hour.

  The subscription fee for an online project management software rg3* qffx 8zrt 2o*58fb jeqj:by AB-Tech is 40 dollars per month. If the customer buys for a whole yq35 j88fretj2 fzg :b**fqo xrear in advance, a discount of 130 dollars is applied.
This can be modelled by the following function, P(n), which gives the total cost when paying annually for the subscription.
P(n)=40n−130,n≥12where n is the number months.
1.Find the total cost of buying a subscription for 2 years. P =    dollars
2.Find $P^{−1}$(1790).    months
The subscription price for a different online project management tool by YZ-tech is 35 dollars per month, however customers can only purchase annually in advance, and there are no discounts. The total subscription cost of YZ-tech's software is less than AB-tech's software when n>k .
3.Find the minimum value of k.   

  The area, A, of a given square can be ;4n3osqb s sm 0+oas2wrepresented by the function
A(P)​=$\frac{P^2}{16}$​,P≥0,​
where P is the perimeter of the square.
The graph of the function 0≤P≤20, is shown below.

1.Find the value of A(20).   
2.On the grid above, draw the graph of the inverse function, $A^{−1}$.
3.In the context of the question, explain the meaning of $A^{−1}$(4)=8.

  The perimeter, P, of a given rdv8f 8ag)os ei(lz n8uixh7 ,6ectangle can be represented by the function
P(A)​=6​$\sqrt{\frac{A}{2}}$,A≥0,​where A is the area of the rectangle.
The graph of the function 0≤A≤24, is shown below.

1.Find the value of P(24). ≈   
2.On the grid above, draw the graph of the inverse function, $P^{−1}$.
3.In the context of the question, explain the meaning of $P^{−1}$(12)=8.

  The circumference, C, of a given circle can be rl/wkuj3ba wy:nx7yz*(9 g5 p yepresented by the function
C(A)​=2π$\sqrt{\frac{A}{π}}$​,A≥0,​where A is the area of the circle.
The graph of the function C, for 0≤A≤24, is shown below.



1.Find the value of C(24). C ­­≈   
2.On the grid above, draw the graph of the inverse function, $C^{−1}$.
3.In the context of the question, explain the meaning of $C^{−1}$(2$π$)=$π$.

A conical funnel is used to add water to a scientific experim)2q +zhdeoqm n o iuiq4w1+)6jent. The funnel initialzq6 )j2o qn+duime4ow1q h)i+ly contains 3 L of water, and the water flows out of the funnel at a rate of 40 mL per minute.


1.Determine a function for the volume V (in millilitres) of water remaining in the funnel after t minutes.
2.If there is V mL of water in the funnel, show that the height of the water can be expressed as
h(V)=$3^3\sqrt{\frac{V}{π}}$​​ cm.
3.Find the composite function $h∘V$, and interpret what it means in the context of this question.
4.Find $(h∘V)(15)$ and explain what it means in the context of this question.

A town is planning to construct a jogging path in a grasr) 2,i5mflp4k ( madogc*w4 uws field 170 m long and 70 m wide. The path is to be the shape of a rectangle with two semicircles of radius x, as shown in the diagram. The sides of the rectangle connecting the a fwrckp4lg,i) ( u*m42w5 modcircles are to be 100 m long.

1.Write down a function, P, (in metres) for the perimeter of the jogging path, in terms of the radius, x.
2.Determine the domain and range of P, taking into consideration the dimensions of the grass field.
3.Find an equation for the inverse function $P^{−1}$(x). Express your answer in the form $P^{−1}$(x)=mx+c.
The designers of the path are deciding whether the total length of the path should be 300 m, 400 m, or 500 m. The designers want to maximise the perimeter of the path, but fit the path in the grass field.
4.Determine which length is most suitable, given the dimensions of the grass field.

Harry is planning on constructing a glass window in one of the outer walls of *wpophspa0b/d.(w5 vf2(xg e his house5vpwwegax hs0p (db./2*(f po . The dimensions of the wall space available are 2m x 2m. Harry wants the window to be in the shape shown in the diagram below. The bottom section is a rectangle and the top part is a semicircle of radius x m. Harry wants the height of the rectangle to be fixed at 1 m.



1.Write down a function P (in metres) for the perimeter of the window in terms of the radius, x.
2.Determine the domain and range of P, taking into consideration the dimensions of the available wall.
3.Find an equation for the inverse function $P^{−1}$(x). Express your answer in the form $P^{−1}$(x)=mx+c.
Harry wants to maximise the size of the window, however the window frame that covers the perimeter of the window can only be 5, 6, or 7 metres long, due to manufacturing restrictions.
4.Determine which perimeter length is the best option for Harry.

Consider the followie +pds i)x uz(9ue;fv-y-m f5zng composite shape, consisting of a rectangle of length 50 cm and varying width x cm, and three-quarter circle with its center at a vertex of the fyvu -z9p)-m dsuxz(e5e+if; rectangle and radius equal to the rectangle width, x cm.



1.Determine a function, L, in terms of x, for the total length of line seen in the diagram. This consists of both the perimeter of the rectangle and circumference of the three-quarter circle.2
Constraints are placed on the dimensions of the composite shape. The total length and width of the composite shape cannot exceed 100 cm and 80 cm respectively.
2.Determine the domain and range of L, taking into consideration these constraints.
3.Find an equation for the inverse function $L^{−1}(x)$. Express your answer in the form $L^{−1}$(x)=mx+c.
Suppose L can only be in multiples of 100 cm (i.e., 100 cm, 200 cm, 300 cm, ...).
4.Determine the maximum length of L that satisfies the constraints.

The radius of a cylindrical containerabxzr0a4 v3irx +q/3l with height x cm is
r(x)=$\frac{56}{x}$​,4≤x≤16​1.Find the range of r.
The function $r^{−1}$ is the inverse function of r.
2.(1)Find $r^{−1}(8).
(2)Interpret the answer to part (b) (i) in context.
(3)Find the range of $r^{−1}$.

A cable making machine in a factory produc6s s:pxzx9 e83wghl*fes 5 metres of cable every 3 minutes. After 4 hours of continuous use, the machine requires 48 minutes of preventative maintenance. Apart from this preventat8ex g:3pszxs6f*w h9live maintenance, the machine works continuously without interruption.
1.Determine the function, $L$, that represents the length of cable produced in terms of time, t, measured in days.
The company sells the cable it produces and has found that the income (in dollars) from selling L metres of cable can be modelled by the function
$I(L)=(3−\sqrt{7}​)L−500​$2.Determine a function for income, I, in terms of time, t, in days.
The company is considering an investment in a new machine that produces 6 metres of cable every 3 minutes and needs 60 minutes of preventive maintenance for every 7 hours of use.
3.Show that the income function for this new machine, in terms of the number of days, t, can be expressed as.
$I_2​(t)=(3−\sqrt{7}​)(2520t)−500​$
4.Determine a function, D, to model the difference in incomes between the two machines, in terms of the number of days, t.
The company decides to purchase the new machine only if it can recover the cost of the machine through the difference in incomes over six month period (assume 180 days).
5.Find the highest amount the company will be willing to pay for the new machine, rounded to the nearest dollar.

  James goes on a cross-country bike ride which comprises of 8 km of flat terraich i58ee r1 *qulne3+an followed by 12 km of constant uphill sloping terrain. We know James can cycle at a speed of 24 km/hr in flat terrain, however we don't know his speed on h e c*eane85+q1rl3uisloping uphill terrain, and therefore denote this by x km/hr.
1.Show his average speed over the entire ride, in terms of x, can be expressed as
S(x)=​​$\frac{60x}{x+36}$
2.Find $S^{−1}$(x).
3.Hence find the speed he needs cycle over the sloping uphill terrain in order to obtain an average speed of 20 km/hr for the whole ride.    km/h

A function is defined by g(x)hh ci+ixbo 630=3-$\frac{12}{x+3}$​ for −9≤x≤9, x≠−3
1.Find the range of g.
2.Find the value of $g^{−1}$(0).

A function is defined by f(xhx)aj:p)ne 8 )izp+l k)=$\sqrt{x+5}$​, x≥−5
1.Determine the domain and range of f.
2.Find the value of $f^{−1}(\sqrt{6}​)$.

A function is defined dx+ :rgey*h9jby f(x)=$\frac{\sqrt{x+1}}{x^2-4}$, for the domain −1≤x≤5, x≠2
1.Find the range of $f$.
2.Find the value of $f^{−1}(\frac{2}{5}​$).

Emily is training for a triathlon and cycles at a speed of 20 km/h)p(twqcc 8r a9r for 15 minutes foc) qrac98p(t wllowed by t hours running at a speed of 15 km/hr.
1.Show that a function to model her average speed S(t) is given by
S(t)= $\frac{20+60t}{1+4t}$​.​
2.Find $S^{−1}(t)$.
3.Hence find the time, t, if the average speed Emily travels at is 16 km/hr.

The function f is definl1kwdrd0i k5;ed by $f(x)=\frac{7x-24}{x-2}$, for −3≤x≤12,x≠2.
1.Find the range of f.
2.Find an expression for the inverse function $f^{−1}(x)$.
3.Write down the range of $f^{−1}$.