{"question":{"question":{"content":"The graph of $y=f(x)$ is shown below. ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/bed2a09b2b8613590418577b8f3d5a945c1e2e95) **Which of the following could be function $f$?** [[☃ radio 1]] ","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/bed2a09b2b8613590418577b8f3d5a945c1e2e95":{"width":425,"height":425},"https://ka-perseus-graphie.s3.amazonaws.com/3da65dc95b1a776a55e113e2eccd07275c918e53.png":{"width":425,"height":425}},"widgets":{"radio 1":{"type":"radio","alignment":"default","static":false,"graded":true,"options":{"choices":[{"content":"$f(x) = \begin{cases} (x+5)^3-1&&\text{if }-7 $ f(x) = \left\{ \begin{array}{lr} \blueD{\text{Equation of graph on left}} &\text{ if left interval} \\ \greenD{\text{Equation of graph on right}} &\text{ if right interval} \end{array} \right.$ ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/1e4d2d5ff30f07e9df4ec65fda602edc881e9eac) An open circle at $(c,d)$ indicates that $c$ *is not included* in the interval associated with that piece, while a closed circle at $(c,d)$ indicates that $c$ *is included* in the interval associated with that piece.","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/a3ecebe83f6b48b63b0f87a4d01f9f1306f523b5":{"width":310,"height":310},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/1e4d2d5ff30f07e9df4ec65fda602edc881e9eac":{"width":310,"height":310}},"widgets":{}},{"content":"###The left-hand piece $\blueD{\text{The graph on the left}}$ is $y=(x+5)^3-1$. [[☃ explanation 1]] It is confined to the interval $-7$\begin{align}g(x)&=x^3\\\\ g(x+5)&=(x+5)^3\\\\ g(x+5)-1&=(x+5)^3-1 \end{align}$ Therefore, it follows that $y=(x+5)^3-1$.","widgets":{}},"version":{"major":0,"minor":0}}}},{"content":"###The right-hand piece $\greenD{\text{The graph on the right}}$ is $y=\sqrt{x}+1$. [[☃ explanation 1]] It is confined to the interval $0\leq x<7$. ","images":{},"widgets":{"explanation 1":{"type":"explanation","alignment":"default","static":false,"graded":true,"options":{"static":false,"showPrompt":"Explain","hidePrompt":"Got it, thanks!","explanation":"Notice that the graph of this function has the basic shape of $y=\sqrt{x}$. ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/36884883ebbb9e415b791f7b0b4dea40834784d1) However, the graph of the function that we are interested in has been moved up $1$ unit. So if $h(x)=\sqrt{x}$, then the function we are interested in is $y=h(x)+1$. This function is built below. >$\begin{align}h(x)&=\sqrt{x}\\\\ h(x)+1&=\sqrt{x}+1\\\\ \end{align}$ Therefore, it follows that $y=\sqrt{x}+1$.","widgets":{}},"version":{"major":0,"minor":0}}}},{"content":"###Conclusion Putting all of this together we get the following piecewise function. $f(x) = \begin{cases} \blueD{ (x+5)^3-1}&&\text{if }-7

Graphs of nonlinear piecewise functions