Question

A function $f:R\to R$ is defined by $f\left(x\right)=x^2$. Determine.
i) Range of f
ii) $\{x:f(x)=4\}$
iii) $\{y:f(y)=-1\}$

Answer 1

We have,
$f\left(x\right)=x^2\dots \dots \left(i\right)$
(i) Clearly range of $f=R^{+}$ (set of all real numbers greater than or equal to zero)

(ii) We have,
$\{x:f(x)=4\}\Rightarrow f\left(x\right)=4\dots \dots \left(ii\right)$
Using equation (i) and equation (ii), we get
$x^2=4\Rightarrow x=\pm 2\therefore \{x:f(x)=4\}=\{-2,3\}$

(iii) $\{y:f(y)=-1\}$
$\Rightarrow f\left(y\right)=-1\dots \dots \left(i\right)$
Clearly, $x^2\ne -1$ or $x^2\ge 0$
$\Rightarrow f\left(y\right)\ne -1$
$\therefore \{y:f(y)=-1\}=\varphi$