Absolute values and the correct inverse branch · source reference

Complete question

For real $x\ne-1/2$, $f(x)=|(1-2x)/(2x+1)|$; for all real x, $g(x)=\tan^{-1}(2x)$. (i) Show f is not one-one. Find the least k such that restricting the domain to $[k,\infty)$ makes it one-one. [2] On this restricted domain, (ii) define the inverse of f. [3] (iii) Find the exact range of $gf^{-1}$. [2]

[2011/AJC/Promo/8]

Printed working

Part (i)

The horizontal line $y=3$ cuts the graph of $f$ at more than one point (two points), so $f$ is not one-one. Least $k=1/2$.

The graph labels vertical asymptote $x=-1/2$, horizontal asymptote $y=1$, cusp $(1/2,0)$ and the counter-example line $y=3$. Its two intersections are at $x=-1$ and $x=-1/4$, both giving3.

Part (ii)

Let $y=|(1-2x)/(2x+1)|$, $x\ge1/2$:

$$\begin{aligned} y&=-\frac{1-2x}{2x+1}\\ &=\frac{2x-1}{2x+1}=1-\frac2{2x+1}\\ \frac2{2x+1}&=1-y\\ x&=\frac12\left(\frac2{1-y}-1\right)=-\frac12+\frac1{1-y}. \end{aligned}$$

Therefore $f^{-1}(x)=-1/2+1/(1-x)$, $D_{f^{-1}}=R_f=[0,1)$.

Thus $f^{-1}:x\mapsto-1/2+1/(1-x)$, $0\le x<1$.

Part (iii)

$$R_{f^{-1}}=D_f=[1/2,\infty)\xrightarrow{g}R_{gf^{-1}}=[\pi/4,\pi/2).$$

Thus $R_{gf^{-1}}=[\pi/4,\pi/2)$. The source arctangent graph labels both horizontal asymptotes $y=\pm\pi/2$, point $(1/2,\pi/4)$, restricted domain ray $R_{f^{-1}}$ beginning at $1/2$ and range with closed lower and open upper endpoints.

Teacher margin notes

For the graphical method, draw horizontal line $y=3$ on the same diagram as the graph of $f$, showing it cuts that graph at two points.

Note: (1) To show $f$ is not1–1, only give an example of one particular horizontal line cutting its graph more than once. (2) Either write “graph of f”, “graph of y=f(x)”, or “the curve y=f(x)”.

The source says for $x\ge1/2$, $1-2x<0$, $2x+1>0$, so $|1-2x|=-(1-2x)$ and $|2x+1|=2x+1$, hence $|(1-2x)/(2x+1)|=|1-2x|/|2x+1|=-(1-2x)/(2x+1)$. At the included endpoint $x=1/2$, the first strict sign should be $\le0$; the displayed absolute-value identity and inverse remain valid there.

Source diagrams

Provenance

Question page references: 3. Solution page references: 12, 13. 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.