Piecewise Functions

[PrinceJ]

I’m trying to do my AP calc summer work but got stuck on this one problem. I have no idea how to define this piecewise function as it is not a normal one. It looks like it possible has two quarter circles but circles are not functions so how do I define it (The rest of the questions are all like this)?

 

[MarkFL]

Hello, and welcome to FMH!

It appears to me we have 3 parts, two circular, and one linear.

a) A circle of radius 1 centered at $(-4,0)$.

b) A circle of radius 3 centered at the origin.

c) A line with passing through $(0,3)$ and $(3,-3)$ which you've already found.

What are the other two parts?

 

[ksdhart2]

The very first thing I'd note is that your "if x = 0" condition is redundant. Specifying that $f(x) = -2x + 3, \: \text{if} \: 0 \leq x \leq 3$ would suffice because $f({\color{red}0}) = -2({\color{red}0}) + 3 = 3$. That aside, I agree with your assessment that the portion of the graph in the upper-left quadrant looks like a quarter circle of radius 3. You (should) know that the equation of such a circle (were it complete) is $x^2 + y^2 = 9$. In order to express this as a function, we want to write it in the form $y = \text{(something)}$. So, how might you go about solving that equation for $y$? What restrictions on $x$ are necessary to ensure it's a quarter circle rather than a half circle?

Finally, the portion of the graph in the lower-left quadrant also resembles a quarter circle. But what is its radius? What would the equation of that full circle be? How might you solve that for $y$? What restrictions on $x$ are necessary to ensure it's a quarter circle rather than a half circle?

 

[pka]

I’m trying to do my AP calc summer work but got stuck on this one problem. I have no idea how to define this piecewise function as it is not a normal one. It looks like it possible has two quarter circles but circles are not functions so how do I define it (The rest of the questions are all like this)?

View attachment 13251

\(\displaystyle f(x)=\begin{cases}\text{a quarter circle, center (-4,-1), radius 1} &, -4\le x\le -3
\\\text{a quarter circle, center (0,0), radius 3} &, -3\le x\le 0\\\text{ line segment } &, 0\le x\le 3 \end{cases}\)

 

[Deleted member 4993]

I’m trying to do my AP calc summer work but got stuck on this one problem. I have no idea how to define this piecewise function as it is not a normal one. It looks like it possible has two quarter circles but circles are not functions so how do I define it (The rest of the questions are all like this)?

View attachment 13251

"....but circles are not functions so how do I define it "

The circular part can be described as a function, if we restrict the domain, like<

$\displaystyle \displaystyle{y = \sqrt{r^2 - x^2}\ \ \ 0 \le x \le r}$

"square-root" operation always returns positive value (by convention).

 

[MarkFL]

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