Question

Which of the function defined below are one-one function(s)?

• A. $f\left(x\right)=x+1,(x\ge -1)$
• B. $g\left(x\right)=x+\frac{1}{x},(x\ge 0)$
• C. $h\left(x\right)=x^2+4x-5,(x>0)$
• D. $f\left(x\right)=e^{-x},(x\ge 0)$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $f\left(x\right)=x+1,(x\ge -1)$
C $h\left(x\right)=x^2+4x-5,(x>0)$
D $f\left(x\right)=e^{-x},(x\ge 0)$

A function is called one-one if it is monotonic
A. $f\left(x\right)=x+1)\Rightarrow f^{\prime }(x)=1>0\Rightarrow f$ is monotonically increasing
B. $g\left(x\right)=x+\frac{1}{x}\Rightarrow g^{\prime }\left(x\right)=1-\frac{1}{x^2},$ which is -ve for $0\le x<1$ and +ve for $x>0\Rightarrow f$ is not monotonic
C. $h\left(x\right)=x^2+4x-5\Rightarrow h^{\prime }\left(x\right)=2x+4>0\forall x>0\Rightarrow f$ is monotonically increasing
D. $f\left(x\right)=e^{-x}\Rightarrow f^{\prime }\left(x\right)=-e^{-x}<0\forall x\le 0\Rightarrow f$ is monotonically decreasing