# How to solve piecewise equation with function of a function

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

• 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 – El borito Jan 22 at 6:42 ## 2 Answers 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 $$2$$. • So does that mean that the whole function is a period function, including the piecewise equation? – mathrook1243 Jan 22 at 6:49 • the whole function? hmmm... we only have one function isn't it. First draw the part from$-1$to$1$and then repeat it. – Siong Thye Goh Jan 22 at 6:50 • so how would I sketch the function between -5 and 5?? – mathrook1243 Jan 22 at 10:06 • First sketch the function from$(-1, 1], then repeat the patterm on the interval $(1,3]$? – Siong Thye Goh Jan 22 at 10:07

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 $$n$$].

• 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. – John Omielan Jan 22 at 7:43
• @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 – Kabo Murphy Jan 22 at 7:47
• 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. – John Omielan Jan 22 at 7:50