To solve this function, we simply need to understand that finding  \(\displaystyle f(10)\) means that \(\displaystyle x=10\) in this specific case. So, we can just substitute 10 in for \(\displaystyle x\).

 \(\displaystyle 0.3(10)^{2}=0.3\cdot 100=30\) 

\(\displaystyle 9x\) is equal to \(\displaystyle 90\), so our final answer is

 \(\displaystyle 30-90\) or \(\displaystyle -60\).

In the relation \(\displaystyle x = y^{2} + 1\), there are many values of \(\displaystyle x\) that can be paired with more than one value of \(\displaystyle y\) - for example, \(\displaystyle (2,1),(2,-1)\).

To demonstrate that  \(\displaystyle y\) is a function of \(\displaystyle x\) in the other examples, we solve each for \(\displaystyle y\):

\(\displaystyle x+y = 1,000,000\) can be rewritten as \(\displaystyle y = 1,000,000 -x\).

\(\displaystyle x^{3}y = -1\) can be rewritten as \(\displaystyle y = -\frac{1}{x^{3}}\)

\(\displaystyle xy = 20\) can be rewritten as \(\displaystyle y = \frac{20}{x}\)

\(\displaystyle y = \left | x-100\right |\) need not be rewritten. 

In each case, we see that for any value of \(\displaystyle x\), \(\displaystyle y\) can be uniquely defined.

\(\displaystyle f(6)= 2(6)^2 +6+2\)\(\displaystyle 2\times ×36+6+2\)\(\displaystyle 72+6+2=80\)

To form this sequence, alternately multiply by 2 and add 5:

\(\displaystyle \begin{matrix} 10 \cdot 2 = 20\\ 20+5 = 25\\ 25 \cdot 2 = 50\\ 50 + 5 = 55\\ 55 \cdot 2 = 110 \\ 110 + 5 = 115 \end{matrix}\)

To keep the pattern going, double the seventh term to get the eighth:

\(\displaystyle 115 \cdot 2 = 230\)

\(\displaystyle f (x) = 3 - \sqrt{x-1}\)

\(\displaystyle f (13) = 3 - \sqrt{13-1} = 3 - \sqrt{12}\)

\(\displaystyle g(x) = 3 + \sqrt{x-1}\)

\(\displaystyle g(13) = 3 + \sqrt{13-1} = 3 + \sqrt{12}\)

The easiest way to find \(\displaystyle \left (fg \right ) (13)\) is to take advantage of the fact that the radical expressons are conjugates, and that their product follows the difference of squares pattern.

\(\displaystyle fg (13) = f (13) \cdot g (13) = (3 - \sqrt{12}) (3 + \sqrt{12})= 3^{2} - \left (\sqrt{12} \right )^{2} = 9- 12 = -3\)

The domain of \(\displaystyle fg\) is the intersection of the domains of the functions \(\displaystyle f\) and \(\displaystyle g\). Both domains are restricted by the same radical expression; since it must hold that the common radicand \(\displaystyle x-1\) is positive:

\(\displaystyle x-1 > 0\) or \(\displaystyle x > 1\)

\(\displaystyle -4\) is therefore outside of the domains of \(\displaystyle f\) and \(\displaystyle g\) and, subsequently, that of \(\displaystyle fg\).

To get each member of this sequence, add a number that increases by one with each element:

\(\displaystyle 7 + 1 = 8\)

\(\displaystyle 8+2 = 10\)

\(\displaystyle 10+3= 13\)

\(\displaystyle 13+4=17\)

\(\displaystyle 17+5 = 22\)

\(\displaystyle 22+6=28\)

To get the next element, add 7:

\(\displaystyle 28 +7 =35\)

Replace \(\displaystyle x\) with \(\displaystyle 3c-8\) in the definition, then simplify.

\(\displaystyle g (x) = 5x - 2\)

\(\displaystyle g (3c-8) = 5(3c-8) - 2 = 5 \cdot3c - 5\cdot 8 -2 = 15c -40-2 = 15c - 42\)

Replace \(\displaystyle x\) with \(\displaystyle 2c-7\) in the definition, then simplify.

\(\displaystyle g (x) = 3x + 2\)

\(\displaystyle g (2c-7) = 3 (2c-7) + 2 = 3 \cdot 2c-3 \cdot 7 + 2= 6c -21+ 2 = 6c-19\)

\(\displaystyle f(-5)= (-5)^4-(-5)^3 +7(-5)-5\)

\(\displaystyle (-5)^4 = (-5)(-5)(-5)(-5)= 5^4=625\)

\(\displaystyle (-5)^3 = (-5)(-5)(-5) = -5^3=-125\)
\(\displaystyle (-5)^4-(-5)^3=625-(-125)=625+125=750\)
\(\displaystyle f(-5)= (-5)^4-(-5)^3 +7(-5)-5\)

\(\displaystyle f(-5)= 750+7(-5)-5\)

\(\displaystyle f(-5)= 750-35-5\)
\(\displaystyle 750-35=715\)
\(\displaystyle 715-5=710\)
\(\displaystyle f(-5) = 710\)