The piece-wise function is defined as follows:
- [tex]f(x) = x^2, 0 \leq x < 3[/tex].
- [tex]f(x) = -\frac{5}{3}x + 14, 3 < x \leq 6[/tex].
- [tex]f(x) = \frac{3}{2}x - 5, 6 \leq x \leq 10[/tex].
What is a piece-wise function?
A piece-wise function is a function that has multiple definitions, depending on the input.
In this graph, for x at least 0 and less than 3, the parabolic curve passes through (0,0), (1,1), (2,4) and has an open interval at (3,9), hence the definition is:
[tex]f(x) = x^2, 0 \leq x < 3[/tex]
For x greater than 3 and at most 6, it is a line going through (3,9) and (6,4), hence:
[tex]m = \frac{9 - 4}{3 - 6} = -\frac{5}{3}[/tex]
[tex]f(x) = -\frac{5}{3}x + b[/tex]
Goes through (3,9), hence:
[tex]9 = -\frac{5}{3}(3) + b[/tex]
b = 14.
So
[tex]f(x) = -\frac{5}{3}x + 14, 3 < x \leq 6[/tex]
For x between 6 and 10, it is a line going through (6,4) and (10,10), hence:
[tex]m = \frac{10 - 4}{10 - 6} = \frac{6}{4} = \frac{3}{2}[/tex]
Then:
[tex]f(x) = \frac{3}{2}x + b[/tex]
When x = 10, f(x) = 10, hence:
[tex]10 = \frac{3}{2}(10) + b[/tex]
10 = 15 + b
b = -5.
Hence:
[tex]f(x) = \frac{3}{2}x - 5, 6 \leq x \leq 10[/tex]
More can be learned about piece-wise functions at https://brainly.com/question/24734454
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