2.2 Functions, domain, range and inverse questions for IB Maths AA

Find the smallest value of $b$ for which the inverse function $f^{-1}$ exists.

For this value of $b$, write down the range of $f$.

For this value of $b$, find an expression for $f^{-1}(x)$.

Show that $\cos \mathrm{C}\widehat{\mathrm{A}}\mathrm{B}={\displaystyle \frac{5x-80}{3x}}$.

Find the value of $x$ for which $\mathrm{C}\widehat{\mathrm{A}}\mathrm{B}=90\mathrm{°}$.

Find the set of possible values of $x$.

Show that the area of the enclosure, $E$ m${}^2$, is given by $E=20\sqrt{-x^2+50x-400}$.

Find the maximum area of the enclosure and the value of $x$ for which it occurs.

Find the values of $x$ for which the area of the enclosure is $200$ m${}^2$.

A ladder of length $5$ m slides so that its foot moves away from the wall at a constant speed of $0.3$ m s${}^{-1}$.

Find the speed at which the top of the ladder moves down the wall when the foot is $3$ m from the wall.

Explain why this model is unrealistic as the top of the ladder approaches the ground.

Consider a ladder of length $L$ m. Let M be the midpoint of [AB].

Show that, for every position of the ladder, M lies on a circle, and state the centre and radius of this circle.

Let $\theta$ be the angle between the ladder and the ground, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that the line (AB) has equation $x\sin \theta +y\cos \theta =L\sin \theta \cos \theta$.

Consider the curve $E$ with equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=L^{\frac{2}{3}}$, for $x>0$, $y>0$.

Use implicit differentiation to show that ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}=-{\left({\displaystyle \frac{y}{x}}\right)}^{\frac{1}{3}}$.

The point P has coordinates $(L{\cos }^3\theta ,L{\sin }^3\theta )$, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that P lies on $E$.

Show that the tangent to $E$ at P is the line (AB) from part (c).

Explain briefly the significance of the result in part (f) for the sliding ladder.

Now consider the family of curves $x^n+y^n=1$, for $x>0$, $y>0$, where $n>0$. The tangent to such a curve at the point $(p,q)$ meets the $x$-axis at X and the $y$-axis at Y.

Show that the tangent at $(p,q)$ has equation $p^{n-1}x+q^{n-1}y=1$.

Hence show that ${\mathrm{X}\mathrm{Y}}^2=p^{2-2n}+q^{2-2n}$.

Show that when $n={\displaystyle \frac{2}{3}}$, the length XY is the same for every point $(p,q)$ on the curve.

By considering the points $({\displaystyle \frac{1}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}})$ and $({\displaystyle \frac{\sqrt{3}}{2}},{\displaystyle \frac{1}{2}})$, show that the length XY is not constant when $n=2$.

Find the value of $k$.

Find the median of $X$. Give your answer in the form $p-q\sqrt{3}$, where $p,q\in {\mathbb{Z}}^{+}$.

Find the value of $a$ and the value of $b$.

Write down the range of $f$.

Find $f^{-1}(x)$, stating its domain.

The graph of $f$, the line $x=b$ and the $x$-axis enclose a region. Find the area of this region.

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.