IB Maths AI SL – Topic 2: Functions

The line $L$ passes through $A(2, 5)$ and $B(6, 13)$.(a) Find the gradient of $L$.(b) Write the equation of $L$ in the form $y = mx + c$.(c) Find the $x$-intercept of $L$.

Let $f(x) = 3x – 4$.(a) Find $f^{-1}(x)$.(b) Find $f^{-1}(11)$.(c) State the domain of $f^{-1}$.

Let $f(x) = x^2 + 1$ and $g(x) = 2x – 3$.(a) Find $f(g(2))$.(b) Find $(g \circ f)(x)$ in simplified form.(c) Solve $(g \circ f)(x) = 15$.

The function $f(x) = -2x^2 + 120x – 1000$ models the daily profit, in dollars, of a stall selling $x$ items.(a) Write down the equation of the axis of symmetry.(b) Find the maximum daily profit.(c) Find the values of $x$ for which the profit is zero.

A population of bacteria is modelled by $N(t) = 5000 \times 1.06^{t}$, where $t$ is the time in hours.(a) Write down the initial population.(b) State the percentage growth rate per hour.(c) Find the time taken for the population to reach $8000$.

A taxi company charges a fixed fee plus a rate per kilometre. A journey of $50$ km costs $\$95$ and a journey of $120$ km costs $\$165$.(a) Find a linear model $C = mk + c$ for the cost $C$ of a journey of $k$ km.(b) Interpret the value of $c$ in context.(c) Find the cost of a $200$ km journey.

A ball is thrown so that its height above the ground, in metres, after $t$ seconds is $h(t) = -5t^2 + 20t + 1.5$.(a) Write down the height from which the ball is thrown.(b) Find the maximum height reached and the time at which it occurs.(c) Find the time at which the ball hits the ground.

A cup of coffee cools according to $T(t) = 20 + 60e^{-0.05t}$, where $T$ is the temperature in $^\circ$C and $t$ is the time in minutes.(a) Write down the initial temperature of the coffee.(b) Find the temperature after $10$ minutes.(c) Find the time taken for the coffee to cool to $30^\circ$C.(d) State the room temperature implied by the model and explain your reasoning.

The amount of a drug in the bloodstream, in mg, $t$ hours after a dose is given by $D(t) = 200 \times 0.8^{t}$.(a) Find the amount remaining after $6$ hours.(b) Find the half-life of the drug in the bloodstream.(c) A second dose must be given once the amount falls below $40$ mg. Find, to the nearest $0.1$ hour, when this happens.

The depth of water $d$ metres in a harbour $t$ hours after midnight is modelled by $d(t) = 3\sin(30t)^\circ + 7$.(a) Write down the maximum and minimum depths.(b) Find the period of the model.(c) Find the depth at $02{:}00$.(d) Find the first time after midnight at which the depth is $10$ m.

A Ferris wheel has diameter $20$ m and its lowest point is $2$ m above the ground. It rotates once every $60$ seconds. A passenger boards at the lowest point at $t = 0$.(a) Write down the height of the centre of the wheel above the ground.(b) Explain why the height $h$ of the passenger after $t$ seconds can be modelled by $h(t) = 12 – 10\cos(6t)^\circ$.(c) Find the height of the passenger after $20$ seconds.(d) Find the length of time during one rotation for which the passenger is more than $17$ m above the ground.

A scientist records the mass $m$ grams of a substance at time $t$ days: $(0, 80)$, $(2, 51.2)$, $(4, 32.8)$, $(6, 21.0)$.(a) Show that an exponential model $m = 80 \times r^{t}$ with $r = 0.8$ fits the data at $t = 2$.(b) Using this model, find the predicted mass at $t = 10$.(c) Use technology to find the exponential regression model $m = ab^{t}$ for the data and state $a$ and $b$ to 3 significant figures.(d) State one limitation of using this model to predict the mass after $50$ days.