can you prove that given function is $x^{-|y|} \ ?$

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Renaming isi1(x, y) as $f(x,y)$ where $x$ is float and $y$ is an integer.

Definition of $f(x,y)$ is given for $x \neq 0 $ as:

\[

f(x,y)=

\begin{cases}

1,& \text{ } y = 0\\

f(x,-y), & \text{} y > 0\\

\frac{f(x,y+1)}{x}, & \text{}y < 0\\

\end{cases}

\]

Case 1: $y > 0$

$f(x, \ y) = f(x, \ -y) = \frac{f(x, \ -y+1)}{x} = \frac{f(x, \ -y+2)}{x^2} = … = \frac{f(x, \ -y+a)}{x^a}$ for $a \geq 1$

So, $f(x,y) = \frac{f(x ,\ a \ – \ y)}{x^a} $

Since, base case is $f(x,0) = 1,$ So, $f(x ,\ a \ – \ y) = 1$ when $a \ – \ y = 0 \Rightarrow a = y$

Hence, $f(x,y) = \frac{1}{x^y} $ for $y > 0$

Case 2: $y < 0$

$f(x, \ y) = \frac{f(x, \ y+1)}{x} = \frac{f(x, \ y+2)}{x^2} = … = \frac{f(x, \ y+a)}{x^a}$ for $a \geq 1$

So, $f(x,y) = \frac{f(x ,\ a \ + \ y)}{x^a} $

Since, base case is $f(x,0) = 1,$ So, $f(x ,\ a \ + \ y) = 1$ when $a \ + \ y = 0 \Rightarrow a = \ - \ y$

Hence, $f(x,y) = \frac{1}{x^ { \ – \ y}} $ for $y < 0$

From case (1) and (2), $f(x,y) = \frac{1}{x^{|y|}} = x^ {\ – \ |y|}$

Hence, given function computes $x^ {\ – \ |y|} \ ; x \neq 0$

Definition of $f(x,y)$ is given for $x \neq 0 $ as:

\[

f(x,y)=

\begin{cases}

1,& \text{ } y = 0\\

f(x,-y), & \text{} y > 0\\

\frac{f(x,y+1)}{x}, & \text{}y < 0\\

\end{cases}

\]

Case 1: $y > 0$

$f(x, \ y) = f(x, \ -y) = \frac{f(x, \ -y+1)}{x} = \frac{f(x, \ -y+2)}{x^2} = … = \frac{f(x, \ -y+a)}{x^a}$ for $a \geq 1$

So, $f(x,y) = \frac{f(x ,\ a \ – \ y)}{x^a} $

Since, base case is $f(x,0) = 1,$ So, $f(x ,\ a \ – \ y) = 1$ when $a \ – \ y = 0 \Rightarrow a = y$

Hence, $f(x,y) = \frac{1}{x^y} $ for $y > 0$

Case 2: $y < 0$

$f(x, \ y) = \frac{f(x, \ y+1)}{x} = \frac{f(x, \ y+2)}{x^2} = … = \frac{f(x, \ y+a)}{x^a}$ for $a \geq 1$

So, $f(x,y) = \frac{f(x ,\ a \ + \ y)}{x^a} $

Since, base case is $f(x,0) = 1,$ So, $f(x ,\ a \ + \ y) = 1$ when $a \ + \ y = 0 \Rightarrow a = \ - \ y$

Hence, $f(x,y) = \frac{1}{x^ { \ – \ y}} $ for $y < 0$

From case (1) and (2), $f(x,y) = \frac{1}{x^{|y|}} = x^ {\ – \ |y|}$

Hence, given function computes $x^ {\ – \ |y|} \ ; x \neq 0$