1. `f(x)=2x+1`, `g(x)=x+5`. Find ` fog(x) `, also evaluate at `x=2`
Solution:
`f(x)=2x+1`
`g(x)=x+5`
`fog(x)=f(g(x))`
`=f(x+5)`
`=2(x+5)+1`
`=2x+11`
`fog(x)=2x+11`
Now find `fog(2)`
`fog(2)``=2*2+11`
`=4+11`
`=15`
`fog(2)=15`
2. `fog(x)=(x+2)/(3x), f(x)=x-2`. Find g(2).
Solution:
`fog(x)=(x+2)/(3x), f(x)=x-2, g(2)=?`
`fog(x)=(x+2)/(3x), f(x)=x-2, g(x)=?`
Let `g(x)=m`
We have `fog(x)=(x+2)/(3x)`
`=>f(g(x))=(x+2)/(3x)`
`=>f(m)=(x+2)/(3x)`
`=>m-2=(x+2)/(3x)`
`=>m-2 = (x+2)/(3x)`
`=>m = (x+2)/(3x)+2`
`=>g(x)=(x+2)/(3x)+2`
`f(x)=x-2, g(x)=(x+2)/(3x)+2, g(2)=?`
Now find `g(2)`
`g(2)``=(2+2)/(3*2)+2`
`=4/6+2`
`=2/3+2`
`=8/3`
`g(2)=8/3`
3. `gof(x)=1/x^2, f(x)=2+x^2`. Find g(x).
Solution:
`gof(x)=1/(x^2), f(x)=2+x^2, g(x)=?`
Let `f(x)=y`
`=>2+x^2=y`
`=>x^2+2 = y`
`=>x^2 = y-2`
`=>x = +- sqrt(y-2)`
The solution is
`x = sqrt(y-2),x = -sqrt(y-2)`
`=>f^(-1)(y)=sqrt(y-2)`
`=>f^(-1)(x)=sqrt(x-2)`
`gof(x)=1/(x^2)`
`=>g(f(f^(-1)(x)))=1/((sqrt(x-2))^2)`
`=>g(x)=1/(x-2)`
This material is intended as a summary. Use your textbook for detail explanation.
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