Is a map $f: x^2 \rightarrow x^2 + \frac{1}{x^2 + 10}$ a linear map?

Question

Is there any way to investigate this question? I'm trying to see whether there are a definite set of shears, scalings, translations, etc., I can do to get from $y=x^2$ to $y=x^2 + \frac{1}{x^2+10}$. In this specific case, from the graph, it seems like some of combination of projection and scaling could do it. Is the method for investigating this dealt with in more advanced linear algebra? All I've had is an intro class. Thank you!

(Oh, and if this isn't a linear map, is it another sort of geometrical map?)

Answer 1

$f(0)=1/10$ so $f$ is not linear.

Answer 2

Use the definition of linearity, $f(ax)\stackrel?=af(x)$:

$$a^2x^2+\frac1{a^2x^2+10}\stackrel?=a\left(x^2+\frac1{x^2+10}\right).$$