The ocean pressure, P, under seg-b vmi:uzvd*x e/06n a level can be modelled by the 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, measuredtw8v3 hge y5 4x8zrtt6 in degrees Celsius, above ground level in Bagan, Mya8tvg38t wxter4z6h5y nmar, can be modelled by the function
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 manufacturingngv, 1. (lv2ejc rl7nn company has found that the number of tyres it produces, N, can be mod(jlrn1nn.ecgv , vl 27elled by 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 softwm+2o2u i sos.h2zvtjz88c( aware by AB-Tech is 40 dollars per month. Ifi8 ozz(mu cs2+s8 j.wt haov22 the customer buys for a whole year 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 re)wntp.wag ; qd ):(z ben./;jqb( ghzzpresented 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 rectangle can be represented by the s5ia)54: e5jwlh meljfunction
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 represented by the functionpn ): hhqh0 ,a;w(esfa
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 experiment. Theov abxm5p(f/f (3;tsh funnel initially contains 3 L of water, and the water flows out of the funnel at/v (;f3saob h5pm(fxt 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 grass field 1 j),q tnjeohc2bd-j3 ;70 m long and 70 m wide. The path is to be the shape of a rectangle with two semicirclesd23)n;eb- jqjoh c,t j of radius x, as shown in the diagram. The sides of the rectangle connecting the circles 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 con2x8mxjqlk )m8structing a glass window in one of the outer walls of his house. The dimensions of the wall space available are 2m x 2m. Harry wants the window to be in the shape shown in the diagram)m2 lm8xq kx8j 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 following composite shape, consisting of a rectangle of length 50f 2 45fdf3p4*7 z.ogjayzzu ik cm and varying width x cm z2ikz 4uzyf adfj3gpf.5o*4 7, and three-quarter circle with its center at a vertex of the 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 container with height x cm keb k6 fftl+/7is
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 facc:wm f6uhs npcq+ 2q* bic4(h1tory produces 5 metres of cable every 3 minutes. After 4 hours ofi6hbhfm4 *w pc+2(s1: ccu nqq continuous use, the machine requires 48 minutes of preventative maintenance. Apart from this preventative 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 gv+oxm 9ye4* iflat terrain 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 sloping uphill terrain, and therefore denote this by x km/m4x+ 9g *yvoeihr.
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)=.sh+/fe1j2vgcgipg8o 8 h y5 t3-$\frac{12}{x+3}$​ 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(x)=w* q3hnv*c -bc$\sqrt{x+5}$​, x≥−5
1.Determine the domain and range of f.
2.Find the value of $f^{−1}(\sqrt{6}​)$.

A function is defined 1rhg* :/wpljc fxm(9 hby 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 spewbwr6c qu0 -4wed of 20 km/hr for 15 minutes followed by t hours running at a speed of 1uq rw0wwc64- b5 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 defined byjozq axxl;)p tf tp*6/7vb7 ,r e:u/fg $f(x)=\frac{7x-24}{x-2}$, 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 $f^{−1}$.