Chapter 29, Problem 1RE

To determine

Whether the given statement is true or false and give explanation.

Answer to Problem 1RE

The given statement is $\underset{\_}{\text{false}}$.

Explanation of Solution

It is given that if the function is $f\left(x,y\right)=\frac{2x^{2}y-y^2}{2xy}$, then $f\left(y^2,x\right)=\frac{2x{y}^4-x^2}{2x^{2}y}$ .

Replace $x=y^2$ and $y=x$ in the function and obtain the value $f\left(y^2,x\right)$ as follows.

$\begin{array}{c}f\left(y^2,x\right)=\frac{2{\left(y^2\right)}^2\left(x\right)-{\left(x\right)}^2}{2\left(y^2\right)\left(x\right)}\\ =\frac{2y^{4}x-x^2}{2y^{2}x}\\ \ne \frac{2x{y}^4-x^2}{2x^{2}y}\end{array}$

That is, $f\left(y^2,x\right)=\frac{2y^{4}x-x^2}{2y^{2}x}$.

The obtained result is not same as the required result as the denominator is not same.

Therefore, the given statement is $\underset{\_}{\text{false}}$.