Answer: 16
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Explanation:
This is a piecewise function. As the name implies, the g(x) is broken up into 3 pieces. Each piece depends on what the input x is.
If x is between [tex]-\infty[/tex] and [tex]-7[/tex], excluding both endpoints, then we pick the first piece. So in this case, [tex]g(x) = x^2-5[/tex]
Or if [tex]-7 \le x \le 2[/tex], then we go for the second piece and [tex]g(x) = 9x-17[/tex]
Lastly, if x is between [tex]2[/tex] and [tex]\infty[/tex], then we go for the last piece and say [tex]g(x) = (x+1)(x-5)[/tex]
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To paraphrase that last section, we have g(x) defined as having a split personality or multiple identities depending on what x is.
- If x is between negative infinity and -7 (exclusive), then g(x) = x^2-5
- If x is between -7 and 2, then g(x) = 9x-17
- If x is between 2 and infinity, then g(x) = (x+1)(x-5)
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The question is: which piece do we pick?
Well, g(7) means that x = 7 for g(x). We'll pick the third piece because 7 is between 2 and infinity. In other words, x = 7 makes [tex]x > 2[/tex] a true inequality.
So,
[tex]g(x) = (x+1)(x-5) \ \text{ when } x > 2\\\\g(7) = (7+1)(7-5) \ \text{ replace every x with 7}\\\\g(7) = (8)(2)\\\\g(7) = 16\\\\[/tex]
