Invert a special point separately · source reference

Complete question

For all real x, $f(x)=\cos x$. For $-1<x\le1$, $g(x)=\ln(1/(1+x))$. Define h by the same logarithmic rule for $-1<x<1$, but set h(1)=−1. (i) Show fg exists. [2] (ii) Expand fg through the cubic term. [4] (iii) State $h^{-1}(-1)$, then define the piecewise inverse as p(x) for x>a and c for x=−1, finding a,c and p. [4]

[2010/MJC/Prelim/I/10]

Printed working

Part (i)

$R_g=[-\ln2,\infty)$ and $D_f=\mathbb R$. Since $R_g\subseteq D_f$, $fg$ exists.

The source graph of $g(x)=-\ln(1+x)$ labels vertical asymptote $x=-1$ and the closed endpoint $(1,-\ln2)$. A composition diagram shows $R_g$ inside $D_f$ and the arrows $g$, $f$ and $fg$.

Part (ii)

$$\begin{aligned} fg(x)&=\cos\left(\ln\frac1{1+x}\right)=\cos(-\ln(1+x))\\ &\approx\cos\left(-x+\frac{x^2}2-\frac{x^3}3\right)\\ &=1-\frac{(-x+x^2/2-x^3/3)^2}{2!}+\frac{(-x+x^2/2-x^3/3)^4}{4!}-\cdots\\ &=1-\frac12\left(-x+\frac{x^2}2\right)^2+\cdots\\ &=1-\frac12(x^2-x^3)+\cdots\\ &=1-\frac12x^2+\frac12x^3+\cdots. \end{aligned}$$

Part (iii)

$h(1)=-1\Rightarrow h^{-1}(-1)=1$.

For the continuous branch, let $y=\ln(1/(1+x))=-\ln(1+x)$, $-1<x<1$:

$$1+x=e^{-y}\Rightarrow x=e^{-y}-1.$$

Thus

$$h^{-1}(x)=\begin{cases}e^{-x}-1,&x>-\ln2,\\1,&x=-1.\end{cases}$$

Teacher margin notes

For $fg$ to exist, the range of the first function $g$ must be a subset of the domain of the second function $f$.

Use the standard Maclaurin series: $\ln(1+x)=x-x^2/2+x^3/3-\cdots+(-1)^{r+1}x^r/r+\cdots$ and $\cos x=1-x^2/2!+x^4/4!-\cdots+(-1)^rx^{2r}/(2r)!+\cdots$.

To find $h^{-1}(x)$: (1) let $y=h(x)\Rightarrow x=h^{-1}(y)$; (2) make $x$ the subject; (3) replace $y$ with $x$; (4) define $h^{-1}$ — both rule and domain. For the branch $-1<x<1$, $D_{h^{-1}}=R_h=R_g\setminus\{-\ln2\}$; the isolated inverse input −1 is handled by the separate case.

Source diagrams

Provenance

Question page references: 3. Solution page references: 13, 14. The original compilation header and printed page numbers are preserved in the edition notes. Original PDFs and page images remain only in the private QA archive.