if $u=x^2$ can you say $x^2$ is a function of $u$

Question

If $$u=x^2$$ then obviously $u$ is a function of $x$ and $x$ is not a function of $u$.

But can we say that $x^2$ is a function of $u$?

Thanks

Answer 1

We can not say that $x$ is a function of $u$. By definition of a function, a function $f$ from a set $X$ to a set $Y$ is defined by a set $G$ of ordered pairs $(x,y)$ with $x\in X$ and $y\in Y$ such that for every $x\in X$, there is only one $y\in Y$. Now, we can see that $x$ is not a function of $u$. We can see that for example $u=4$ has two different $x$ values, namely $x=2$ and $x=-2$. This definition is equivalent with saying $u:\mathbb{R}\to\mathbb{R};x\to x^2$ is not a bijection. If we define $u:[0,\infty)\to[0,\infty);x\to x^2$, we can inverse $u$ such that we can say that $x$ is a function of $u$.

Answer 2

The answer is yes. It is a simple exercise in substitution. Do $t = x^2$ and you see that:

$$t = u$$

And you see $t$ is a function of $u$, actually it is probably the simplest function one can think of. It is like $f(x) = x$ in first precalc course.

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One example where it is common to mention functions of other functions like this is the trigonometric polynomials which due to the trigonometric laws have special properties. For example:

$$f(x)=\sin(x)^2 - 2\sin(x)+1$$

Which we can see that after doing the substitution $$t=\sin(x)$$ It becomes a polynomial:

$$t^2 - 2t + 1$$

Just as the name suggests.