Question:
Show that the function $f: R \rightarrow R: f(x)=x^{5}$ is one $-$ one and onto.
Solution:
To show: $f: R \rightarrow R:: f(x)=x^{5}$ is one - one and onto.
Proof:
$f(x)=x^{5}$
$\Rightarrow y=x^{5}$
Since the lines do not cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)$ is one - one.
The range of $f(x)=(-\infty, \infty)=R$ (Codomain)
$\therefore f(x)$ is onto
Hence, showed $f: R \rightarrow R: f(x)=x^{5}$ is one - one and onto.
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