Problem 4
<h3>Answer:</h3>
![f(x) = \begin{cases}2x+4 \text{ if } x < 1\\x-3 \text{ if } x \ge 1\end{cases}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cbegin%7Bcases%7D2x%2B4%20%5Ctext%7B%20if%20%7D%20x%20%3C%201%5C%5Cx-3%20%5Ctext%7B%20if%20%7D%20x%20%5Cge%201%5Cend%7Bcases%7D)
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Work Shown:
The left line goes through (-2,0) and (0,4)
The slope of this line is
m = (y2-y1)/(x2-x1)
m = (4-0)/(0-(-2))
m = (4-0)/(0+2)
m = 4/2
m = 2
The y intercept is b = 4
Since m = 2 and b = 4, this means y = mx+b turns into y = 2x+4. This portion is only done when x < 1. Note the open circle at the endpoint of this portion. So we do not include x = 1 as part of this piece.
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The line on the right side goes through (1,-2) and (2,-1)
Slope
m = (y2-y1)/(x2-x1)
m = (-1-(-2))/(2-1)
m = (-1+2)/(2-1)
m = 1/1
m = 1
The y intercept is b = -3. You can see this if you extend the line until it crosses the y axis.
Alternatively, plug in (x,y) = (1,-2) and m = 1 into y = mx+b to find that b = -3
So y = mx+b turns into y = 1x+(-3) or just y = x-3
We combine both parts to end up with ![f(x) = \begin{cases}2x+4 \text{ if } x < 1\\x-3 \text{ if } x \ge 1\end{cases}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cbegin%7Bcases%7D2x%2B4%20%5Ctext%7B%20if%20%7D%20x%20%3C%201%5C%5Cx-3%20%5Ctext%7B%20if%20%7D%20x%20%5Cge%201%5Cend%7Bcases%7D)
This is only graphed when
(note the closed or filled in circle for the endpoint of this portion).
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Problem 5
Answer:
<h3>
![f(x) = \frac{1}{2}|x+3|](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%7Cx%2B3%7C)
is the absolute value function</h3><h3>while this is the piecewise function</h3>
![f(x) = \begin{cases}-\frac{1}{2}(x+3) \text{ if } x < -3\\\frac{1}{2}(x+3) \text{ if } x \ge -3\end{cases}\\](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cbegin%7Bcases%7D-%5Cfrac%7B1%7D%7B2%7D%28x%2B3%29%20%5Ctext%7B%20if%20%7D%20x%20%3C%20-3%5C%5C%5Cfrac%7B1%7D%7B2%7D%28x%2B3%29%20%5Ctext%7B%20if%20%7D%20x%20%5Cge%20-3%5Cend%7Bcases%7D%5C%5C)
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Work Shown:
y = |x| .... parent function
y = |x+3| ... shift 3 units to the left
y = (1/2)*|x+3| .... vertically compress by factor of 1/2
f(x) = (1/2)*|x+3|
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Break that down into a piecewise function
when x < -3, then y = -(1/2)(x+3)
when
, then y = (1/2)(x+3)
I'm using the rule that y = |x| turns into y = -x when x < 0 and y = x when
So that is how we get
as the piecewise function.