The definition of the function is (a) [tex]\left[\begin{array}{cc}2x&if\ x < 2\\\frac12 x + \frac 52&if x \ge 1\end{array}\right[/tex]
How to determine the function definition?
The graph shows a piecewise function that has the following domain
x < 1 and x ≥ 1
At x < 1, we have the following points
(0,0) and (1,2)
The linear equation is calculated using:
[tex]y = \frac{y_2 -y_1}{x_2 - x_1} * (x -x_1) + y_1[/tex]
This gives
[tex]y = \frac{2-0}{1 - 0} * (x -0) + 0[/tex]
y = 2x.
So, we have:
y = 2x if x < 1
At x ≥ 1, we have the following points
(1,3) and (3,4)
The linear equation is calculated using:
[tex]y = \frac{y_2 -y_1}{x_2 - x_1} * (x -x_1) + y_1[/tex]
This gives
[tex]y = \frac{4-3}{3 - 1} * (x -1) + 3[/tex]
[tex]y = \frac{1}{2} * (x -1) + 3[/tex]
Expand
[tex]y = \frac{1}{2}x + \frac 52[/tex]
So, we have:
[tex]y = \frac{1}{2}x + \frac 52[/tex] if x ≥ 1
Hence, the definition of the function is (a) [tex]\left[\begin{array}{cc}2x&if\ x < 2\\\frac12 x + \frac 52&if x \ge 1\end{array}\right[/tex]
Read more about piecewise functions at:
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