2.7 Composite and inverse functions questions for IB Maths AI

Find $(g\circ f)(2000)$.

Find an expression for $(g\circ f)(x)$, giving your answer in the form $ax+b$.

Ingrid needs 900 CHF for her stay in Zurich. Find the number of Canadian dollars she must convert. Give your answer correct to two decimal places.

Akira converts 800 000 JPY. Find the number of Australian dollars he receives.

Find an expression for $f^{-1}(x)$.

Find $f^{-1}(5000)$ and interpret your answer in context.

Akira deposits 8154.90 AUD in a savings account that pays a nominal annual interest rate of 3.8%, compounded quarterly. Find the value of his savings after 4 years. Give your answer correct to two decimal places.

The value of Akira's savings, in AUD, after $t$ years is $g(t)=8154.90{\left(1.0095\right)}^{4t}$.

At any time, Akira can convert Australian dollars back to yen at 1 AUD = 97.5 JPY, less a 1% commission, so that $h(x)=0.99\times 97.5x$ is the number of yen he receives for $x$ AUD.

Find $(h\circ g)(4)$, giving your answer to the nearest yen.

Find the number of complete years Akira must leave his money in the account before it can be converted into more than 1 000 000 JPY.

Dr Okafor wants to plot the data. Suggest why she should use a logarithmic scale for the mass.

Find the values of ${\log }_{10}M$ and ${\log }_{10}H$ for the horse.

Let $X={\log }_{10}M$ and $Y={\log }_{10}H$.

Find Pearson's product-moment correlation coefficient, $r$, for $X$ and $Y$. Hence comment on whether a model of the form $H=k{M}^b$ is suitable.

Find the equation of the regression line of $Y$ on $X$.

Use your answer to part (d) to show that $H=k{M}^b$, and find the value of $k$ and the value of $b$.

A giraffe has a mass of 800 kg and a measured resting heart rate of 48 beats per minute.

Use your model to estimate the resting heart rate of the giraffe, and find the percentage error in this estimate.

Find the percentage decrease in resting heart rate that this model predicts when body mass is doubled.

Find the mass of a mammal with a predicted resting heart rate of 150 beats per minute.

Dr Okafor models the mean lifespan, $L$ years, of a mammal by $L=8M^{0.2}$. She investigates the total number of heartbeats, $N$, in a mammal's lifetime, assuming that the heart beats at its resting rate throughout. A year has 525 600 minutes.

Show that $N\approx 1.02\times {10}^9\times M^{-0.058}$.

Find the value of $N$ for a mouse and for an elephant. Give your answers in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A popular science article claims that "every mammal gets about one billion heartbeats in its lifetime". Comment on this claim, with reference to your answers to parts (i)(i) and (i)(ii).

A blue whale has a mass of about 150 000 kg. State one reason why Dr Okafor's model may not give a reliable estimate of its heart rate.

Find $(f\circ g)(3)$.

Find an expression for $(g\circ f)(x)$. Give your answer in the form $a{x}^2+bx+c$.

Solve the equation $(f\circ g)(x)=(g\circ f)(x)$.

Find the radius of the slick 5 hours after the accident.

Find an expression for $(A\circ r)(t)$.

Find the time at which the area of the slick reaches 3000 m².

Find an expression for $r^{-1}(x)$.

Find $r^{-1}(30)$ and interpret your answer in context.

Explain why $r^{-1}(45)$ has no meaning in this context.

Find the rate at which the area of the slick is increasing 5 hours after the accident.

Write down the temperature of the coffee when it is poured.

Find an expression for $T^{-1}(x)$.

Find $T^{-1}(50)$ and interpret your answer in context.

State the domain of $T^{-1}$.