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I have got the question below for an assignment, I understand how and what the piecewise equation does, but I am wondering if someone can explain what the $f(x)=f(x+2)$ is referring to? Does this mean function of a function?If so, how would I solve this? Question

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    $\begingroup$ You can write f(x)=f(x+2) with \$, like this: \$ f(x)=f(x+2)\$ $\Big(f(x)=f(x+2)\Big)$. See the help or here for more information $\endgroup$ – El borito Jan 22 at 6:42
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It means if you know $f(0)$, we know $f(2)$, and we also know $f(4)$. The values are equal.

Similarly for $f(-2)$ and $f(-4)$.

In general, if we know $f(x)$, then we know the values of $f(x+2)$ and $f(x-2)$ and they are equal.

This is a periodic function with period $2$.

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  • $\begingroup$ So does that mean that the whole function is a period function, including the piecewise equation? $\endgroup$ – mathrook1243 Jan 22 at 6:49
  • $\begingroup$ the whole function? hmmm... we only have one function isn't it. First draw the part from $-1$ to $1$ and then repeat it. $\endgroup$ – Siong Thye Goh Jan 22 at 6:50
  • $\begingroup$ so how would I sketch the function between -5 and 5?? $\endgroup$ – mathrook1243 Jan 22 at 10:06
  • $\begingroup$ First sketch the function from $(-1, 1], then repeat the patterm on the interval $(1,3]$? $\endgroup$ – Siong Thye Goh Jan 22 at 10:07
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The function is first defined on $[-1,1]$ and then extended to the whole line by adding the condition $f(x+2)=f(x)$ (periodocity of $f$). For example, if $7 \leq x \leq 9$ then $x-8$ lies between $-1$ and $1$ and $f(x)$ is defined to be $f(x-8x)$. [Note that $f(x+2)=f(x)$ for all $x$ implies that $f(x+2n)=f(x)$ for all $x$ and for any integer $n$].

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  • $\begingroup$ I believe your $2$ uses of $x - 8x$ should be just $x - 8$. Also, at the end, the statement $f\left(x + 2n\right) = f\left(x\right)$ is true for any integers $n$, not just positive ones, as Siong Thye Goh's answer indicates. $\endgroup$ – John Omielan Jan 22 at 7:43
  • $\begingroup$ @JohnOmielan There was a typo. I meant $x-8$. As far as your second comment is concerned the statement '$f(x+2n)=f(x)$ for all $x$ and all positive integers $n$ ' is equivalent to '$f(x+2n)=f(x)$ for all $x$ and all integers $n$ '. Perhaps I could have said 'all integers' for ease of understanding $\endgroup$ – Kabo Murphy Jan 22 at 7:47
  • $\begingroup$ Thanks for correcting the first instance. However, you still have a second one, as I said earlier, with it being in the statement "... is defined to be $f\left(x - 8x\right)$". As for your suggestion of removing the term "positive", although it's not incorrect, I believe it's more general and it will provide ease of understanding. $\endgroup$ – John Omielan Jan 22 at 7:50

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