Question

Graph the following piecewise function on a separate piece of graph paper and upload your graph below.

Answer 1

For this case we have the following functions:
 For $- 4 \leq x \ \textless \ 3$ : 
 $y = x + 4$
 For $3 \leq x \ \textless \ 6$ :
 $y = 2x - 1$
 What we need to know for this case is:
 Both functions are one lines
 Both functions have a positive slope
 Both functions are in a certain interval
 Answer:
 See attached image to see functions

Answer 2

ANSWER

To graph the function

$f(x)=\left \{ {{x+4\:\:if\:\:-4\leq x<3}  \atop {2x-1\:\:if\:\:3\leq x<6}} \right.$

follow the steps below.

1. Find y- intercept by plugging in $x=0$.

$x=0$ is on the interval,  $-4\leq x<3$, so we substitute in to

$f(x)=x+4$

$\Rightarrow f(0)=0+4$

$\Rightarrow f(0)=4$

Hence the y-intercept is $(0,4)$

2. Find x-intercept by setting $f(x)=0$

This implies that

$x+4=0,$ on $-4\leq x<3$

or

$2x-1=0$ on $3\leq x <6$

We now solve for $x$ on each interval,

$x=-4,$ on $-4\leq x<3$

or

$x=\frac{1}{2}$ on $3\leq x <6$

But observe that

$x=\frac{1}{2}$ does not belong to $3\leq x <6$

This means it  can never be an intercept for this piece-wise function.

Hence our x-intercept is $(-4,0)$

3. Plotting the boundaries of the interval.

For $f(x)=x+4$ on  $-4\leq x<3$

$f(-4)=-4+4$

$\Rightarrow f(-4)=0$.

This point $(-4,0)$ coincides with the x-intercept.

$f(3)=3+4$

$f(3)=7$

So we have the point $(3,7)$. But note that $x=3$ does not belong to this interval so we plot this point as a hole.

For $f(x)=2x-1$ on $3\leq x <6$

$f(3)=2(3)-1$

$\Rightarrow f(3)=5$

So we plot $(3,5)$

$f(6)=2(6)-1$

$\Rightarrow f(6)=11$

So we plot $(6,11)$ also as a hole.

Plotting all these points we can now graph the function,

$f(x)=\left \{ {{x+4\:\:if\:\:-4\leq x<3}  \atop {2x-1\:\:if\:\:3\leq x<6}} \right.$

See attachment for graph.