Question

Which of the following functions are not injective map
(A) $f\left(x\right)=x+\frac{1}{x};x\in (0,\infty )$
(B) $g\left(x\right)=x^2+4x-5;x\in (0,\infty )$

Answer 1

(A) Given $f\left(x\right)=x+\frac{1}{x}$; $x\in (0,\infty )$.

Now for $x=2$ and $x=\frac{1}{2}$, we have the same $f-$image $\frac{5}{2}$.

So for two distinct element we have the same image under $f$,

So $f\left(x\right)$ is not injective.

(B) To check the injectivity of $f\left(x\right)=x^2+4x-5$ for $x\in (0,\infty )$, let

$f\left(x_1\right)=f\left(x_2\right)$

or, $x_1^2+4x_1-5=x_2^2+4x_2-5$

or, $(x_1-x_2)(x_1+x_2+4)=0$

As $x_1,x_2\in (0,\infty )\Rightarrow x_1+x_2\ne -4$, so $x_1=x_2$.

So, for $f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$.

So $f\left(x\right)$ is injective.