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Answers:</h3>
![f(g(x)) = \sqrt{x^2+5}+5\\\\g(f(x)) = x+30+10\sqrt{x-1}](https://tex.z-dn.net/?f=f%28g%28x%29%29%20%3D%20%5Csqrt%7Bx%5E2%2B5%7D%2B5%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20x%2B30%2B10%5Csqrt%7Bx-1%7D)
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Work Shown:
Part 1
![f(x) = \sqrt{x-1}+5\\\\f(g(x)) = \sqrt{g(x)-1}+5\\\\f(g(x)) = \sqrt{x^2+6-1}+5\\\\f(g(x)) = \sqrt{x^2+5}+5\\\\](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Csqrt%7Bx-1%7D%2B5%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%20%5Csqrt%7Bg%28x%29-1%7D%2B5%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%20%5Csqrt%7Bx%5E2%2B6-1%7D%2B5%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%20%5Csqrt%7Bx%5E2%2B5%7D%2B5%5C%5C%5C%5C)
Notice how I replaced every x with g(x) in step 2. Then I plugged in g(x) = x^2+6 and simplified.
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Part 2
![g(x) = x^2+6\\\\g(f(x)) = \left(f(x)\right)^2+6\\\\g(f(x)) = \left(\sqrt{x-1}+5\right)^2+6\\\\g(f(x)) = \left(\sqrt{x-1}\right)^2+2*5*\sqrt{x-1}+\left(5\right)^2+6\\\\g(f(x)) = x-1+10\sqrt{x-1}+25+6\\\\g(f(x)) = x+30+10\sqrt{x-1}\\\\](https://tex.z-dn.net/?f=g%28x%29%20%3D%20x%5E2%2B6%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20%5Cleft%28f%28x%29%5Cright%29%5E2%2B6%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20%5Cleft%28%5Csqrt%7Bx-1%7D%2B5%5Cright%29%5E2%2B6%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20%5Cleft%28%5Csqrt%7Bx-1%7D%5Cright%29%5E2%2B2%2A5%2A%5Csqrt%7Bx-1%7D%2B%5Cleft%285%5Cright%29%5E2%2B6%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20x-1%2B10%5Csqrt%7Bx-1%7D%2B25%2B6%5C%5C%5C%5Cg%28f%28x%29%29%20%3D%20x%2B30%2B10%5Csqrt%7Bx-1%7D%5C%5C%5C%5C)
In step 4, I used the rule (a+b)^2 = a^2+2ab+b^2
In this case, a = sqrt(x-1) and b = 5.
You could also use the box method as a visual way to expand out ![\left(\sqrt{x-1}+5\right)^2](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%7Bx-1%7D%2B5%5Cright%29%5E2)
Answer:
C
Step-by-step explanation:
The general rule is that any 2 must exceed the third in order for a triangle to form. If any one is an exception, it won't work.
A will form a triangle. Any 2 exceeds the third.
B also forms a triangle. Same rule.
C does not work. 7 + 11 = 18 which is the value of the third leg.
D words Any two exceeds the 3rd.
(40+x)/2=60
40 +x = 120
x = 80
check :
40+80 = 120
120/2 = 60
so his second round needs to be 80 mph