Let $D=R-\{0,1\}$ and $f: D \rightarrow D, g: D \rightarrow D$ and $h: D \rightarrow D$ be three functions…
Let $D=R-\{0,1\}$ and $f: D \rightarrow D, g: D \rightarrow D$ and $h: D \rightarrow D$ be three functions defined by $f(x)=\frac{1}{x} ; g(x)=1-x$ and $h(x)=\frac{1}{1-x}$. If $j: D \rightarrow D$ is such that (gojof) $(x)=f(x)$ for all $x \in D$, then which one of the following is $j(x)$ ?
$(f \circ g)(x)$
$f(x)$
$g(x)$
$(g \circ h)(x)$
Solution
We have,
$\begin{aligned}
& (\text { goj of })(x)=f(x) \text { for all } x \in D \\
& \Rightarrow \quad g((j o f)(x))=\frac{1}{x} \\
& \Rightarrow \quad 1-(j o f)(x)=\frac{1}{x} \\
& \Rightarrow \quad 1-j(f(x))=\frac{1}{x} \\
& \Rightarrow \quad 1-j\left(\frac{1}{x}\right)=\frac{1}{x} \\
& \Rightarrow \quad j\left(\frac{1}{x}\right)=1-\frac{1}{x} \\
& \Rightarrow \quad j(x)=1-x=g(x) \\
\end{aligned}$