Plugging in 2 wherever $\displaystyle x$ is present in the formula yields an answer of 14.

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.

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

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