# For the given functions f , g, and h, find `fogoh` and state the exact domain of `fogoh` . Please show all of your work.`f(x)=sqrt(x)` `g(x)=e^x-(11e^x)/2+24` `h(x)=2x`

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Expert Answers

rcmath | Certified Educator

`fogoh(x)=f(g(2x))=f(e^(2x)-[11e^(2x)]/2+24)=`

`sqrt(e^(2x)-[11e^(2x)]/2+24) =>`

`fogoh(x)=sqrt((-9)/2e^(2x)+24)```

For this function to be defined, you need the value under the square root to be non-negative, thus

`(-9/2)e^(2x)+24>=0 =>`

`(-9/2)e^(2x)>=-24 =>`

`e^(2x)<=(-24)*(-2/9) =>`

`e^(2x)<=16/3 =>`

`2x<=ln(16/3) =>`

`x<=(1/2)ln(16/3)`

So the domain is `(-oo,(1/2)ln(16/3)]`