- Suppose I am asked to find the zeros of :
$$g(x)= - \frac 12 (\frac 19x^2 - 4)^2$$.
- Is the following reasoning formally correct:
(1) $g(x)= f(\frac 13 x)$ with $f(x)= - \frac 12 (x^2 - 4)^2$.
(2) I know that in general, if $g(x)=f(kx)$ then $g = \{(\frac xk, y) | (x,y)\in f\}$.
(3) The set of all points in $f$ with a null $y$ cordinate is : $\{ (-2, 0) , (2,0) \}$ , since
$ - \frac 12 (x^2 - 4)^2 = 0 \iff (x^2 - 4)^2 = 0 \iff (x^2 - 4)=0 \iff x^2 = 4$ $ \iff (x=2 \lor x=-2)$.
(4) So, the set of all points in $g$ with a null $y$-coordinate is :
$\{(\frac {-2}{k}, 0), (\frac {2}{k}, 0) \} = \{ ( \frac {-2}{\frac 13}, 0), (\frac {2}{\frac 13}, 0)\} = \{( -6,0), (6,0)\}$
(5) Therefore, the zeros of function $g$ are $x=-6$ and $x=6$.
My question : is using transformation facts a valid way of solving an equation or of finding the zeros of a function? Incidentally, is it standardly accepted in an exam?
Yes, but why in this case?
$$g(x) = 0 \Leftrightarrow -\frac{1}{2}(\frac{1}{9}x^2-4)(\frac{1}{9}x^2-4)=0 \Leftrightarrow (\frac{1}{9}x^2-4)=0 \Leftrightarrow x^2 = 36 \Leftrightarrow x = \pm 6.$$