Further Functions & Graphs (DP IB Analysis & Approaches (AA): SL): Exam Questions

The function $f$ is defined by $f\left(x\right)=\frac{2x+1}{x-4}$, where $x\in \mathrm{ℝ}$, $x\ne 4$.

Write down the equation of

(i) the vertical asymptote of the graph of $f$

(ii) the horizontal asymptote of the graph of $f$.

Find $f^{-1}\left(x\right)$, the inverse function of $f\left(x\right)$.

Write down the equation of the vertical asymptote of the graph of $y=f^{-1}\left(x\right)$.

Let $f\left(x\right)=\ln \left(x+2\right)$, for $x>-2$.

Find the coordinates of the point where the graph of $f$ intersects

(i) the $x$-axis

(ii) the $y$-axis.

State the equation of the vertical asymptote to the graph of $y=f\left(x\right)$.

The graph of $y=f\left(x\right)$ intersects the graph of $y=f^{-1}\left(x\right)$ at two points.

Find the coordinates of these two points.

Let $f\left(x\right)=0.5{\mathrm{e}}^{2x}+1$, for $-1\le x\le 2$.

On the following grid, sketch the graph of $y=f\left(x\right)$.

The inverse function of $f$ can be written as $f^{-1}\left(x\right)=A\ln \left(b\left(x-c\right)\right)$.

Find the value of $A$, the value of $b$ and the value of $c$.

Carbon-14 is a radioactive isotope of carbon. It decays exponentially, losing mass as it does so, and it is used in carbon dating to estimate the age of objects.

The time it takes for the mass of carbon-14 to halve is called its half-life. The half-life of carbon-14 is approximately 5700 years.

The mass, $m$ grams, of carbon-14 in an object $t$ years after it was formed can be modelled by

$m=m_0{\mathrm{e}}^{-kt}$

where $m_0$ and $k$ are constants.

An object initially contains 100 grams of carbon-14.

Write down the value of $m_0$.

Explain why $m=50$ when $t=5700$.

Show that $k=1.22\times {10}^{-4}$, correct to three significant figures.

A different object currently contains 60 grams of carbon-14.

Find the mass of carbon-14 that remains in this object after a further 2000 years.

A small company makes a profit of £2500 in its first year of business and £3700 in its second year. The company's profit, $P$ pounds, in year $n$ of its business can be modelled by

$P=P_0{n}^k$

where $P_0$ and $k$ are constants.

Write down two equations connecting $P_0$ and $k$.

Find the value of $P_0$ and the value of $k$.

Use the model to predict the company's profit in its third year and in its fourth year.

Show that $P=P_0{n}^k$ can be written as $\log P=\log P_0+k\log n$.

To help prevent extinction, scientists released some rare birds into a new nature reserve.

The number of birds, $B$, in the reserve $t$ years after the release can be modelled by

$B=16{\mathrm{e}}^{0.85t}$

Write down the number of birds the scientists released into the reserve.

Use the model to find the number of birds in the reserve after 3 years.

Find the time, in years, for the number of birds in the reserve to reach 500.

Rebecca recently had the COVID-19 vaccine. The amount of vaccine, $V_1$ mg, in her bloodstream $t$ days after 9 am on Monday can be modelled by

$V_1\left(t\right)=1.7t{\mathrm{e}}^{-1.25t},t\ge 0$

On the following grid, sketch the graph of $y=V_1\left(t\right)$.

Find, to the nearest minute, the time and the day on which $V_1$ reaches its maximum value.

Rebecca experienced side effects from the time when the amount of vaccine reached its maximum value until the amount had dropped to half of its maximum value.

Find, to the nearest minute, the length of time for which Rebecca experienced side effects.

The vaccine is considered to have left Rebecca's bloodstream once the amount drops to 1% of its maximum value.

Find the time $t$ at which this happens.

Rebecca's friend, Zara, had the vaccine at the same time. The amount of vaccine, $V_2$ mg, in Zara's bloodstream can be modelled by

$V_2\left(t\right)=1.766t{\mathrm{e}}^{-1.3t},t\ge 0$

Find, to the nearest minute, how much sooner $V_2$ reaches its maximum value than $V_1$.

Let $f\left(x\right)=-\frac{3}{x-3}$, for $x\ne 3$.

(i) Write down the coordinates of the $y$-intercept of the graph of $f$.

(ii) Explain why the graph of $f$ has no $x$-intercept.

(iii) Write down the range of $f$.

Find the value of $f^{-1}(-1).$

Given that $g\left(x\right)=f\left(x+3\right)+1$, find the domain and the range of $g$.

Finn borrows $3200 from his parents and decides to pay them back $c$ dollars in the first month and then $m$ dollars each subsequent month.

After two months Finn has paid back his parents a total of $1000, this can be expressed as $m+c=1000$. After half a year he still owes his parents $1000.

Find another equation connecting $m$ and $c$.

Find the value of m and c.

Finn's parents add a charge of 6.25% of the $3200 to the amount he owes.

Calculate the number of months it takes Finn to pay back his parents.

Let $f\left(x\right)=\frac{7}{2\left(x-7\right)}-5$, for $x\ne 7$.

Find the coordinates of the point where the graph of $f$ crosses

(i) the $x$-axis

(ii) the $y$-axis.

Write down the equation of

(i) the vertical asymptote of the graph of $f$

(ii) the horizontal asymptote of the graph of $f$.

Let $g\left(x\right)=2\left(1-2x\right)$, for $x\in \mathrm{ℝ}$. The graphs of $f$ and $g$ intersect at the points $\mathrm{P}$ and $\mathrm{Q}$.

Write down the coordinates of $\mathrm{P}$ and of $\mathrm{Q}$.

Find $\mathrm{PQ}$.

A function is defined by $f\left(x\right)=4-\frac{12}{5x+9}$, for $x\in \mathrm{ℝ}$, $x\ne a$.

Write down the value of $a$.

Find the range of $f$.

Find the value of $f^{-1}\left(2\right)$.

The average fat-free mass, $M$ kg, of a footballer aged $a$ years can be modelled by

$M\left(a\right)=10\log \left(a-15\right)+50,16\le a\le 25$

Find the average fat-free mass of footballers aged

(i) 16 years

(ii) 25 years.

A linear model, $L\left(a\right)=ma+c$, gives the same values as $M$ at $a=16$ and at $a=25$.

Find the value of $m$ and the value of $c$.

The number of bacteria, $n$, in a dish $t$ minutes after the start of an experiment can be modelled by

$n=5231{\mathrm{e}}^{0.12t}$

Write down the initial number of bacteria.

Find the number of bacteria after 12 minutes. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

Find the value of $t$ when $n=2.7\times {10}^4$.

Let $f\left(x\right)=2x^4-2x^3-4x^2+x+1$, for $x\in \mathrm{ℝ}$.

Find the zeros of $f$. Hence, solve $f\left(x\right)<0$.

For the graph of $f$, find the coordinates of

(i) the local maximum point

(ii) the two local minimum points.

The intensity of light, $I$, is assumed to be 100% at the surface of the ocean and decreases with depth, $d$. It can be estimated by the function

$I\left(d\right)=k{\left(1.08\right)}^{-d}$

where $I$ is expressed as a percentage, $d$ is the depth below the surface, in metres, and $k$ is a constant.

Write down the value of $k$.

Write down the domain and the range of $I$.

Find the intensity of light 6.2 m below the surface.