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g(x)=3-1/3=8/3 f(x)=8/3+1/8/3=73/24 73+24=97

g(3) = 3 - 1/3 = 8/3 f(g(3)) = 8/3 - 1/8/3 = 8/3 - 3/8 = 73/24 = a/b a= 73, b = 24 a+b = 97

first, find the solution of g(x), g(x)= g(3)=x-1/x = 3-1/3 = (9-1)/3 = 8/3 ................(1) Now, putting value of g(x) on fucntion f(x) i.e. putting 8/3 for finding f(g(3)) = x + 1/x = 8/3 +3/8 = (64+9)/24 = 73/24. therefor, f(g(3)) = 73/24 ................(2) As given in problem f(g(x)) =a/b .................(3) comparing equation (2) & (3) a/b = 73/24 therefore, a+b = 73+24 = 97. answear

g(3)=3-1/3=8/3 f(g(3))=8/3-3/8=73/24 a=73 b=24 a+b=73+24=97

g(3) = 8/3 f(g((3))= 73/24 73+24=97

f(g(x))=(x-(1/x))+1/(x-(1/x)) if x=3 then f(g(3))=73/24 a=73&b=24,as they are coprime integers. Hence,a+b=73+24=97

$g({x})={x}-\frac{1}{x}$

$g({3})={3}-\frac{1}{3}$

which gives $\frac{8}{3}$

$f({x})={x}+\frac{1}{x}$

$f(g({3}))=\frac{8}{3}+\frac{3}{8}$

which simplifies to $\frac{73}{24}$

But $\frac{a}{b}+\frac{73}{24}$

Therefore ${a}+{b}={73}+{24}$

$\boxed{97}$

g(3)=8/3

f(8/3)=73/24

73+24=97

$g(3)=3-\frac{1}{3}=\frac{8}{3}$ . $f(g(3))=f(\frac{8}{3})=\frac{8}{3}+\frac{3}{8}=\frac{73}{24}$ . Answer is $a+b=73+24=\boxed{97}$

Solution - First of all, evaluate g(3) . To do this, all we have to do is to put the value 3 in the function: $g(x)= x - \frac {1}{x}$ $g(3)= (3) - \frac {(1)}{(3)}$ $g(3)= \frac {((3 \times 3) - 1)}{3}$ $g(3)= \frac {(9-1)}{3}$ $g(3)= \frac {8}{3}$

Now, the value of g(3) serves as argument for the function f(x) , and hence we evaluate the value of f(g(3) : $f(g(3)) = \frac {8}{3} + \frac {1}{\frac {8}{3}}$ $f(g(3)) = \frac {8}{3} + \frac {3}{8}$ $f(g(3)) = \frac {(8 \times 8) + ( 3 \times 3)}{3 \times 8}$ $f(g(3)) = \frac {64+9}{24}$ $f(g(3)) = \frac {73}{24}$

Now, since 73 and 24 are co-primes , & hence satisfy the given condition for variables a and b . We have, a= 73; b=24 . Finally, calculating a+b , we get: $a+b= 73+24= 97$ So, required solution is 97 .