Question 1200349
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Answers:
f(-5)= <font color=red size=4>-20</font>
f(-4)= <font color=red size=4>-2</font>
f(4)= <font color=red size=4>6</font>
f(5)= <font color=red size=4>7</font>
f(8)= <font color=red size=4>9</font>



Explanation:


Piecewise functions are a Frankenstein of sorts because they have multiple functions glued together.


Another way to look at it is that the piecewise function has split identities.


The input x will determine which piece to use.<ul><li>If {{{x < -4}}}, then {{{f(x) = 4x}}}</li><li>If {{{-4 <= x <= 5}}}, then {{{f(x) = x+2}}}</li><li>If {{{x > 5}}}, then {{{f(x) = (x-5)^2}}}</li></ul>If the input is x = -5, then we go for the interval {{{x < -4}}}. Draw out a number line to see why this is the case.
This means we pick the first piece.
{{{f(x) = 4x}}}
{{{f(-5) = 4(-5)}}}
{{{f(-5) = -20}}}



If the input is x = -4, then we go for the interval {{{-4 <= x <= 5}}}
This time we go for the 2nd piece.
{{{f(x) = x+2}}}
{{{f(-4) = -4+2}}}
{{{f(-4) = -2}}}


We stick with the 2nd piece for the input x = 4, since it is also in the interval {{{-4 <= x <= 5}}}
{{{f(x) = x+2}}}
{{{f(4) = 4+2}}}
{{{f(4) = 6}}}


Same goes for x = 5
{{{f(x) = x+2}}}
{{{f(5) = 5+2}}}
{{{f(5) = 7}}}


The last input x = 8 will involve the third piece, since we're now in the interval {{{x > 5}}}
{{{f(x) = (x-5)^2}}}
{{{f(8) = (8-5)^2}}}
{{{f(8) = (3)^2}}}
{{{f(8) = 9}}}



This is what the piecewise graph looks like
{{{drawing(400,400,-12,17,-34,38,
graph(400,400,-12,17,-34,38,
4x*((sqrt(-(x+4.1)))/(sqrt(-(x+4.1)))),
(x+2)*((sqrt(-(x+4)(x-5)))/(sqrt(-(x+4)(x-5)))),
(x-5)^2*((sqrt(x-5.3))/(sqrt(x-5.3)))),

red(circle(-4,-16,0.5)),

green(circle(-4,-2,0.1)),
green(circle(-4,-2,0.2)),
green(circle(-4,-2,0.3)),
green(circle(-4,-2,0.4)),
green(circle(-4,-2,0.5)),


green(circle(5,7,0.1)),
green(circle(5,7,0.2)),
green(circle(5,7,0.3)),
green(circle(5,7,0.4)),
green(circle(5,7,0.5)),

blue(circle(5,0,0.5))

)}}}
The first piece 4x is in red. It is graphed only when {{{x < -4}}}
The second piece x+2 is in green. It is graphed only when {{{-4 <= x <= 5}}}
The third piece (x-5)^2 is in blue. It is graphed only when {{{x > 5}}}


There are open holes at (-4, -16) and (5, 0)
The open holes are not part of the graph. Think of them as potholes in the road.
There are closed filled in endpoints at (-4, -2) and (5, 7)


I recommend either Desmos or GeoGebra as two graphing options.
Here's the link to the interactive Desmos graph
<a href = "https://www.desmos.com/calculator/jkfomczjuz">https://www.desmos.com/calculator/jkfomczjuz</a>
This will allow you to zoom in/out, move the window around, etc.
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