Question

please help out with this integration question​

Answer 1

Question :

If f (x) = $\frac{3x+5}{7x+2}$ and g (x) = x² + 7, find fog (3), fog (-3), gof (2) and fof (-3)

Solution :

Given, f (x) = $\frac{3x+5}{7x+2}$

g (x) = x² + 7

So, g (3) = 3² + 7 = 9 + 7 = 16

Now, fog (3)

= f (g (3))

= f (16)

= $\frac{3(16)+5}{7(16)+2}$

= $\frac{48+5}{112+2}$

= $\frac{53}{114}$

Also, g (-3) = (-3)² + 7 = 9 + 7 = 16

Since, g (3) = g (-3), we can say that

fog (-3) = fog (3) = $\frac{53}{114}$

Now, f (2)

= $\frac{3(2)+5}{7(2)+2}$

= $\frac{6+5}{14+2}$

= $\frac{11}{16}$

Now, gof (2)

= g (f (2))

= g ($\frac{11}{16}$)

= $(\frac{11}{16})^{2}+7$

= $\frac{121}{256}+7$

= $\frac{121+1792}{256}$

= $\frac{1913}{256}$

Now, f (-3)

= $\frac{3 (-3)+5}{7 (-3)+2}$

= $\frac{-9+5}{-21+2}$

= $\frac{-4}{-19}$

= $\frac{4}{19}$

So, fof (-3)

= f (f (-3))

= f ($\frac{4}{19}$)

= $\dfrac{3(\frac{4}{19})+5}{7(\frac{4}{19})+2}$

= $\dfrac{\frac{12}{19}+5}{\frac{28}{19}+2}$

= $\frac{12+95}{28+38}$

= $\frac{107}{66}$

Answer 2

Answer: fog(-3)  = 107/66 and gof(2)  = 1913/256

Step-by-step explanation:

We have,

f(x) = (3x+5)/(7x+2

g(x) = x² +7

Now,

f(-3) = [3*(-3)+5]/[7*(-3)+2]

       = (-9+5)/(-21+2)

       = -4/-19

       = 4/19

$f(\frac{4}{19})=\frac{3*\frac{4}{19}+5}{7*\frac{4}{19}+2}\\\;\\f(\frac{4}{19})=\frac{\frac{12}{19}+5}{\frac{28}{19}+2}\\\;\\f(\frac{4}{19})=\frac{\frac{12+95}{19}}{\frac{28+38}{19}}\\\;\\f(\frac{4}{19})=\frac{107}{66}$

fof(-3) = f(f(-3))

           = f(4/19)

fof(-3)  = 107/66

f(2) = [3*2+5]/[7*2+2]

      = (6+5)/(14+2)

      = 11/16

g(11/16) = (11/16)² +1

            = 121/256 + 7

             = (121+1792)/256

             = 1913/256

gof(2) = g(f(2))

          = g(11/16)

gof(2)  = 1913/256