Q. 11 - Calculus

Question

The functions $f\left(x\right)=2^{\left(x^2\right)}$ and $g\left(x\right)={\left(2^x\right)}^2$look similar, but are very different functions. (This is why we try to avoid using the ambiguous notation $2^{x^2}$.) Calculate  $f\left(3\right)$ and $g\left(3\right),$ and show that they are not the same (and thus that $f\left(x\right)$ and $g\left(x\right)$ are not the same function). Then find all the values for which $f\left(x\right)=g\left(x\right).$

Step-by-Step Solution

Answer

$f\left(3\right)=512$ and $g\left(3\right)=64$ and $f\left(3\right)$ and $g\left(3\right)$ are not the same. $f\left(x\right)=g\left(x\right)$ at $x=0$ and $x=2.$

Step 1. Given information

It is given that $f\left(x\right)=2^{\left(x^2\right)}$ and $g\left(x\right)={\left(2^x\right)}^2$ and it looks similar. We need to calculate $f\left(3\right)$ and $g\left(3\right)$ and show that they are not the same. Also, we need to determine the value for which $f\left(x\right)=g\left(x\right).$

Step 2. Calculation

$f\left(x\right)=2^{\left(x^2\right)}.$

$f\left(3\right)=2^{\left(3^2\right)}.$

        $=2^9.$

        $=512.$

$g\left(x\right)={\left(2^x\right)}^2.$

 $g\left(3\right)={\left(2^3\right)}^2.$

         $=8^2.$

          $=64.$

Thus, $f\left(3\right)$ and $g\left(3\right)$ are not the same.

Step 3. Solution

$f\left(x\right)=g\left(x\right).$

$2^{\left(x^2\right)}={\left(2^x\right)}^2.$

$2^{x^2}=2^{2x}.$

$x^2=2x.$

$x^2-2x=0.$

$x\left(x-2\right)=0.$

$x=0 or x-2=0.$

                     $x=2.$

Thus, $f\left(x\right)=g\left(x\right)$ at $x=0$ and $x=2.$