Identity rules still have domains · source reference

Complete question

(a) If f is one-one, determine with justification whether $ff^{-1}$ exists. [1] (b) $g(x)=\ln x$ for real $x\ge1$; $h(x)=x^2-2x+2$ for real $x>0$. (i) Given gh exists, define it including its domain and range. [2] (ii) Sketch g, its inverse, and $g^{-1}g$ together. [3] (iii) Find the inputs satisfying $g^{-1}g(x)=gg^{-1}(x)$. [1]

[2011/DHS/Prelim/I/4]

Printed working

Part (a)

$f$ is one-one, so $f^{-1}$ exists. Since $R_{f^{-1}}=D_f\subseteq D_f$, the composite function $ff^{-1}$ exists.

Part (b)(i)

$D_{gh}=D_h$. Thus $gh:x\mapsto\ln(x^2-2x+2)$, $x>0$.

$R_h=[1,\infty)$; the source sketch of $h$ labels minimum $(1,1)$ and an open point at $(0,2)$. The graph of $g$ starts at the closed point $(1,0)$.

$$D_h=(0,\infty)\xrightarrow{h}R_h=[1,\infty)\xrightarrow{g}R_{gh}=[0,\infty).$$

Thus $R_{gh}=[0,\infty)$. The accompanying composition diagram places $R_h$ inside $D_g$ and arrows $D_h\xrightarrow{h}R_h\xrightarrow{g}R_{gh}$, with an outer composite arrow $gh$.

Part (b)(ii)

On equal-scale axes, draw $y=g(x)=\ln x$, $x\ge1$, starting at $(1,0)$; $y=g^{-1}(x)=e^x$, $x\ge0$, starting at $(0,1)$; and $y=g^{-1}g(x)=x$, $x\ge1$, starting at $(1,1)$. The underlying reflection line $y=x$ is dashed where it is not part of the restricted composite.

Part (b)(iii)

$$g^{-1}g(x)=gg^{-1}(x)=x.$$

Thus $x\in D_g\cap D_{g^{-1}}$, so $x\in[1,\infty)$.

Teacher margin notes

$gh(x)=g(x^2-2x+2)$. To find $R_{gh}$: (1) find $R_h$ using the graph of the first function $h$; (2) sketch the second function $g$; (3) $R_h$ becomes the restricted domain of $g$; (4) use $R_h$ as the new domain and the graph of $g$ to find its range, which is $R_{gh}$.

When sketching $f$ and $f^{-1}$ on the same diagram, the scale must be the same for the x- and y-axes; their graphs must be symmetrical about $y=x$. Always include endpoints. $D_{g^{-1}g}=D_g=[1,\infty)$.

$D_{gg^{-1}}=D_{g^{-1}}=[0,\infty)$. To compare the two composites, $x\in D_{g^{-1}g}\cap D_{gg^{-1}}\Rightarrow x\in D_g\cap D_{g^{-1}}$.

Source diagrams

Provenance

Question page references: 2. Solution page references: 10. 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.