Question

The graph of the function `y=f(x)` is as shown in the figure.

Then draw the graphs of
(i) `|y|=sgn(f(x))` (ii) `|y|=|f(x|)|`
(iii) `y=x^(sgn)(f(x))`

Answer 1

(i) `|y|=sgn(f(x))`
`f(x)gt0if-2ltxlt1andxgt3implies|y|=1`
`f(x)lt0if1ltxlt3implies|y|=-1"(not possible")`
At `x=-2,1,3,f(x)=0implies|y|=0`
Hence the graph is as follows.

(ii) `|y|=|f(|x|)|`
Graph of `y=f(|x|)`

Graph of `y=|f(|x|)|`

Graph of `|y|=|f(|x|)| `

(iii) `y=x^(sgnf(x)` For `f(x)gt0,-2ltxlt1orxgt3impliesy=x`
For `f(x)lt0,1ltxlt3impliesy=(1)/(x)`
For `x=1,3,sgn(x)impliesy=1`
Hence the graph of the function is as shown in the following figure.