Substitute \(\displaystyle f(x) = 9 -x^3\) into the function \(\displaystyle g(x) = ln (x+1)\) for \(\displaystyle x\).

Then it will become:

\(\displaystyle g(9 - x^3) = ln ((9-x^3) + 1) = ln (10 - x^3)\)

f(g(x)) simply means: where ever you see an x in the equation f(x), replace it with g(x).

So, doing just that, we get 

\(\displaystyle 3-2(e^x)^2+\ln(e^x)\),

which simplifies to 

\(\displaystyle 3-2e^{2x}+x\ln(e)\).

Since 

\(\displaystyle \ln(e)=1\) our simplified expression becomes,

\(\displaystyle 3-2e^{2x} +x\).

g(f(x)) simply means replacing every x in g(x) with f(x).

\(\displaystyle \frac{3(2-5x)^2}{(2-5x)}-6\)

After simplifying, it becomes

\(\displaystyle \frac{3(2-5x)(2-5x)}{(2-5x)}-6\)

\(\displaystyle =3(2-5x)-6\)

\(\displaystyle =6-15x-6=-15x\)

The composite function means to plug in the function of \(\displaystyle g(x)\) into the function \(\displaystyle f(x)\) for every x value in the function.

Therefore the composition function becomes:

\(\displaystyle f\circ g = f[g(x)] = 4(x^2) = 4x^2\).

The composite function means to plug in the function \(\displaystyle g(x)\) into \(\displaystyle f(x)\) for every x value.

Therefore the composite function becomes,

\(\displaystyle f\circ g = f[g(x)] = (x^2)+4 = x^2+4\)

When doing a composition of functions such as this one, you must always remember to start with the innermost parentheses and work backward towards the outside.

So, to begin, we have

 \(\displaystyle h(-7)=(-7)+8=1\).

Now we move outward, getting 

\(\displaystyle g(h(-7))=g(1)=6(1)+4=10\).

Finally, we move outward one more time, getting 

\(\displaystyle f(g(h(-7)))=f(g(1))=f(10)=(10)^2-3=97\).

Solve for the value of \(\displaystyle i(2)\).

\(\displaystyle i(2)= 2^4=16\)

Solve for the value of \(\displaystyle h(i(2))\).

\(\displaystyle h(i(2))=h(16) = 3\)

Solve for the value \(\displaystyle g(h(i(2)))\).

\(\displaystyle g(h(i(2)))=g(3) = 3\)

The composite function notation \(\displaystyle f\circ g\) means to swap the function \(\displaystyle g(x)\) into \(\displaystyle f(x)\) for every value of \(\displaystyle x\). Therefore:

\(\displaystyle f\circ g\)

\(\displaystyle =f[g(x)]\)

\(\displaystyle =(x^{3})+7\)

\(\displaystyle =x^{3}+7\)

The composite function notation \(\displaystyle g\circ f\) means to swap the function \(\displaystyle f(x)\) into \(\displaystyle g(x)\) for every value of \(\displaystyle x\). Therefore:

\(\displaystyle g\circ f\)

\(\displaystyle =g[f(x)]\)

\(\displaystyle =4(x-7)+6\)

\(\displaystyle =4x-28+6\)

\(\displaystyle =4x-22\)

The composite function notation \(\displaystyle f\circ g\) means to swap the function \(\displaystyle g(x)\) into \(\displaystyle f(x)\) for every value of \(\displaystyle x\). Therefore:

\(\displaystyle f\circ g\)

\(\displaystyle =f[g(x)]\)

\(\displaystyle =(2x+3)^{2}-4\)

\(\displaystyle =4x^{2}+12x+9-4\)

\(\displaystyle =4x^{2}+12x+5\)