The function $f(x)=x^2$ is scale-invariant because $f(\lambda x)=\lambda^2f(x)$ but $g(x)=x^3+x^2$ is not because $g(\lambda x)\neq\lambda^{\Delta}g(x)$.
How can we understand this feature by comparing the graphical plots of $f(x)$ and $g(x)$?
Graphical understanding of why the function $g(x)=x^3+x^2$ is not scale invariant unlike $f(x)=x^2$
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functions
polynomials
graphing-functions
1 Answers
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If $f$ is a function satisfying $f(tx) = t^{2} f(x)$ for all real $t$, and if some point $(x_{0}, y_{0})$ with $x_{0} \neq 0$ lies on the graph, then $(tx_{0}, t^{2}y_{0})$ lies on the graph for all real $t$. This implies $$ f(x) = \frac{y_{0}}{x_{0}^{2}}\, x^{2}. $$ Particularly, the graph is either the $x$-axis (if $y_{0} = 0$), or a parabola with vertex at the origin (if $y_{0} \neq 0$).
If $r$ is a real number, the same idea shows that a function $g$ satisfying $g(x_{0}) = y_{0}$ for some $x_{0} > 0$, and $g(tx) = t^{r}g(x)$ for all real $t > 0$, has the form $$ g(x) = \frac{y_{0}}{x_{0}^{r}}\, x^{r},\quad x > 0. \tag{*} $$
For every real $r$, the function $g(x) = x^{3} + x^{2}$ is not of the form (*).