2.5 Composite functions questions for IB Maths AA

Show that $\cos (\arcsin x)=\sqrt{1-x^2}$.

Hence find the exact value of $\sin (2\arcsin \frac{1}{3})$.

Solve the equation $\cos (\arcsin x)=x$.

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

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

Solve the equation $(g\circ f)(x)=2$, for $0\le x\le \pi$.

Solve the equation $(f\circ g)(x)=(g\circ f)(x)$, for $0\le x\le \pi$.

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.

For $z=3+\text{i}$, find $f(z)$ and $g(z)$.

Describe fully the geometric transformation represented by $f$ and the geometric transformation represented by $g$.

Let $z=x+y\text{i}$, where $x,y\in \mathbb{R}$. Show that $(g\circ f)(z)=y+x\text{i}$, and hence describe fully the single geometric transformation represented by $g\circ f$.

The function $T$ is defined by $T(z)=\text{i}{z}^{\ast }+2+2\text{i}$, for $z\in \mathbb{C}$.

Show that there is no complex number $z$ such that $T(z)=z$.

Find $T(T(z))$ in its simplest form, and describe fully the geometric transformation it represents.

Show that, for every $z\in \mathbb{C}$, the midpoint of the points representing $z$ and $T(z)$ lies on the line $\text{Im}(z)=\text{Re}(z)$.

Find the amount of yen received for €$250$. Give your answer to the nearest yen.

Find $Y^{-1}(y)$.

Find $Y^{-1}(20000)$ and interpret its meaning in this context.

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

Write down the range of $f$.

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

Write down the domain of $f^{-1}$.

Solve $f^{-1}(x)=3$.