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Do the graphes of $f(x^2)=\sin(x^2)$ and $g(x)=\sin(x^2)$ look the same?

If this is true then: $f(x^2)=\sin(x^2)=g(x)$ and thus the $x^2$ in the parenthesis of $f$ does not matter at all???

user50224
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    Again, you need to be specific about what you mean. Writing $f(x^2)$ is ambiguous as to whether you mean to graph $y=f(x)$ subject to the constraint $f(x^2)=\sin(x^2)$ or just $\sin(x^2)$. – Adam Hughes Aug 05 '14 at 20:18
  • @AdamHughes I know how to graph $g(x)$. My problem is how to graph $f(x^2)$ because in the brackets of $f$ there is this $x^2$ BUT my coordinate axis is only $x$ My real problem is how to master this difference between $x^2$ and the $x$-axis. I would highly appreciate your help to finally solve my problem :). – user50224 Aug 05 '14 at 20:28
  • @AdamHughes Graphing $g(x)$ is no problem since the variable of $g$ is $x$ and I got an $x$-axis. But what shall I do with the $x^2$ in $f(x^2)$? Shall I set $f(3)$==>$x=\sqrt(3)$ or $f(3^2) on the point of 3 on the x-axis? – user50224 Aug 05 '14 at 20:33
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    $f(x^2) = \sin x^2$ is the same as $f(x) = \sin x, x \ge 0$. So, no, they have different plots. – Kaster Aug 05 '14 at 20:33
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    The problem is that there is no answer any of us can give you that's a "this is it" answer, because the way you've phrased it makes it ambiguous. It's like saying "Did you see the girl with the telescope?" Are you asking if I used a telescope to see the girl in question or are you asking if I saw the girl in question, and this girl had a telescope at the time? The question itself could mean two things the way you've written it, and the answer is different depending on how someone interprets it. – Adam Hughes Aug 05 '14 at 20:35
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    I think you are confused by function notation. You must specify what the independent variable is. When you write "$f(x^2)=\sin(x^2)$" it suggests you are intending the function that maps $\square\mapsto\sin(\square)$. I don't think that's what you really mean. – MPW Aug 05 '14 at 20:46

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This is the same problem you had here.

In the first case, letting $v=x^2\geqslant 0$, the graph you're talking about is described by $\forall v\in\mathbb{R}^+\, (v,f(v))=(v,\sin(v))$.

In the second case, you consider the graph described by $\forall u\in\mathbb{R},\ (u,g(u))=(u,\sin(u^2))$.

Of course $u$ and $v$ can be change to any variable, it is just the name of elements in $\mathbb{R}$. So, you're comparing $(x,\sin(x))$ and $(x,\sin(x^2))$ which are of course not the same, assuming you're plotting as functions of $x$ (and not $x^2$).

But if you plot $(x^2,\sin(x^2))$, then you'll get something on the same curve that $(x,\sin(x))$ (but only in the right half-plane).

Community
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anderstood
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