Show that the function f : R → 3 → R - 2 given by f(x)= x-2 x-3 i… - Vidyadip

Question

Show that the function f : R → {3} → R - {2} given by $\text{f(x)}=\frac{\text{x}-2}{\text{x}-3}$ is a bijection.

✓

Answer

We have, f : R⁺ → R⁺ given by
f(x) = x²
g : R⁺ → R⁺ given by
$\text{g(x)}=\sqrt{\text{x}}$
$\therefore$ fog(x) = f(g(x))
$=\text{f}(\sqrt{\text{x}})=(\sqrt{\text{x}})^2=\text{x}$
Also, gof(x) = g(f(x))
$=\text{g}(\text{x}^2)=\sqrt{\text{x}^2}=\text{x}$
Thus,
fog(x) = gof(x)

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