Can I move a constant out of a continuous function? When?

Question

If I have $f(\frac{1}{k}g(x))$

Then is this (ever) same as

If I have $\frac{1}{k}f(g(x))$?

Is $\sin 2x = 2\sin x$? Here, $f(x) = \sin x, g(x) = x, k=\frac 12$. — Deepak, Dec 04 '20 at 11:00

score 2 · Accepted Answer · answered Dec 04 '20 at 10:58

2

Try with $g(x) = x$ and $f(x) = x^2$.

The property of $f$ that decides whether you can do that or not, is homogeneity. For example, linear functions, such as $f(x) = 21 x$ are homogeneous (of degree one), but $f(x) = x^2$ is not.

answered Dec 04 '20 at 10:58

Antoine

3,439

Good answer that refreshes homogeneity. – mavavilj Jan 17 '21 at 09:02

score 1 · Answer 2 · answered Dec 04 '20 at 11:00

1

No. For example take $f(x)=x^2$ and $g(x)=x.$

Then

$$f(\frac{1}{k}g(x))=\frac{1}{k^2}x,$$

but

$$\frac{1}{k}f(g(x))=\frac{1}{k}x.$$

answered Dec 04 '20 at 11:00

Fred

77,394

But are there exceptions, when the property would hold? – mavavilj Dec 04 '20 at 18:05

Can I move a constant out of a continuous function? When?

2 Answers2