Answers:
- f(14) = 4
- (g o f)(14) = 5
- [tex]g^{-1}(x) = \frac{x+7}{3}[/tex]
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Work Shown:
Plug x = 14 into f(x)
[tex]f(x) = \sqrt{x+2}\\\\f(14) = \sqrt{14+2}\\\\f(14) = \sqrt{16}\\\\f(14) = 4\\\\[/tex]
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Recall that (g o f)(x) is the same as g( f(x) )
This means (g o f)(14) is the same as g( f(14) )
Earlier we found f(14) = 4, so g( f(14) ) = g(4)
Now plug x = 4 into g(x)
[tex]g(x) = 3x-7\\\\g(4) = 3*4-7\\\\g(4) = 12-7\\\\g(4) = 5\\\\[/tex]
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To find the inverse, we swap x and y and solve y like so...
[tex]g(x) = 3x-7\\\\y = 3x-7\\\\x = 3y-7 \ \ \text{ ... swap x and y}\\\\x+7 = 3y\\\\3y = x+7\\\\y = \frac{x+7}{3}\\\\g^{-1}(x) = \frac{x+7}{3}\\\\[/tex]
