# Let f be the function from R to R defined by f(x) = x^{2}. Find

Let f be the function from R to R defined by $f\left(x\right)={x}^{2}$. Find
${f}^{-1}\left(x\mid x>4\right)$
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Given: $f:\mathbf{R}\to \mathbf{R}$
$f\left(x\right)={x}^{2}$
${f}^{-1}\left(\left\{x\mid x>4\right\}\right)$ contains all elements of $\mathbf{R}$ that has as image a real number greater than 4.
$f\left(x\right)>4$
Since $f\left(x\right)={x}^{2}$
${x}^{2}>4$
The square ${x}^{2}$ is more than 4 if x is more than 2 or x is less than -2(since ${2}^{2}=4={\left(-2\right)}^{2}\right).$
$\left(x<-2\right)\vee \left(x>2\right)$
${f}^{-1}\left(\left\{x\mid x>4\right\}\right)$ thus contains all real numbers smaller than -2 and larger than 2.
${f}^{-1}\left(\left\{x\mid x>4\right\}\right)=\left\{x\mid \left(x<-2\right)\vee \left(x>2\right)\right\}$