2.8 Reciprocal and rational functions questions for IB Maths AA

Find the probability that Ravi answers a question correctly.

Given that Ravi answers a question correctly, find the probability that he knew the answer.

Ravi answers two questions. His performances on the two questions are independent. Given that he answers both questions correctly, find the probability that he guessed at least one of the answers.

Show that the probability that Ravi knew the answer, given that he answered correctly, is ${\displaystyle \frac{kp}{kp+1-p}}$.

For the case $p=\frac{3}{5}$, find the least value of $k$ for which the probability that Ravi knew the answer, given that he answered correctly, is at least 0.95.

For a fixed value of $p$, describe what happens to the probability in part (d) as $k$ increases without limit. Give a reason for your answer in the context of the quiz.

Write down the equations of the vertical asymptotes of the graph of $y=f(x)$.

Find the range of $f$ for $\frac{\pi }{4}<x<\frac{3\pi }{4}$.

Solve the equation $f(x)=x$.

Consider the linear function $f(x)=mx+c$, where $x\in \mathbb{R}$ and $m,c\in \mathbb{R}$.

Show that $(f\circ f)(x)=m^{2}x+c(m+1)$.

Hence find all pairs of values of $m$ and $c$ for which $f$ is self-inverse.

Describe the graphs of the self-inverse linear functions found in part (a)(ii).

Now consider $f(x)={\displaystyle \frac{ax+b}{cx+d}}$, where $x\in \mathbb{R},x\mathrm{\ne }-{\displaystyle \frac{d}{c}}$, and $a,b,c,d\in \mathbb{R}$ with $c\mathrm{\ne }0$ and $ad-bc\mathrm{\ne }0$.

Show that $(f\circ f)(x)={\displaystyle \frac{(a^2+bc)x+b(a+d)}{c(a+d)x+bc+d^2}}$.

Hence show that $f$ is self-inverse if and only if $d=-a$.

Write down the value of $p$ for which $h(x)={\displaystyle \frac{px-3}{4x+5}}$ is self-inverse.

For this value of $p$, find the coordinates of the points where the graph of $h$ meets the line $y=x$.

Explain why the graph of a self-inverse function is symmetric about the line $y=x$.

Let $n$ be an odd positive integer and $k\in \mathbb{R}$. Show that $f(x)=\sqrt[n]{k-x^n}$, where $x\in \mathbb{R}$, is self-inverse.

Let $f$ be a self-inverse function, defined for all $x\in \mathbb{R}$, that is strictly increasing.

Prove by contradiction that $f(x)=x$ for all $x\in \mathbb{R}$.

Let $g$ be a self-inverse function, defined for all $x\in \mathbb{R}$, that is strictly decreasing.

Show that the equation $g(x)=x$ has at most one solution.

Write down the value of ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }f(x)}$.

Find the value of $f^{\mathrm{\prime }}(1)$.

Let $g(x)=2f(x)+x$. Find the equation of the tangent to the graph of $g$ at the point where $x=1$.

Show that the probability that the person has the disease, given a positive result, is ${\displaystyle \frac{0.94p}{0.91p+0.03}}$.

Find $f(0.01)$.

Show that $f$ is an increasing function.

Find the value of $p$ for which $f(p)=0.5$.

Find the set of values of $p$ for which a person with a positive result has a probability greater than 0.9 of having the disease.

Find the probability that a person who has positive results in both tests has the disease.

Show that the gradient of the chord $[\mathrm{P}\mathrm{Q}]$ is $-{\displaystyle \frac{4}{a(a+h)}}$.

Hence write down the gradient of the tangent to the curve at P.

The tangent to the curve at P meets the $x$-axis at M and the $y$-axis at N. O is the origin.

Show that the area of triangle OMN does not depend on the value of $a$.