A) many-one and onto
B) many-one and into
C) one-one and onto
D) one-one and into
Correct Answer: D
Solution :
[d] \[f(x)={{e}^{{{x}^{3}}-3x+2}}\] Let \[g(x)={{x}^{3}}-3x+2;\] \[g'(x)=3{{x}^{2}}-3=3({{x}^{2}}-1)\] \[\ge 0\]for \[x\in (-\infty ,-1]\] Therefore, \[f(x)\] is increasing function hence \[f(x)\] is one-one. Now, the range of \[f(x)\] is \[(0,{{e}^{4}}]\]. But co-domain is \[(0,{{e}^{5}}]\]. Hence.,\[f(x)\] is an into function.
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