### Instruction: Consider the given functions: $$\begin{array}{ccc}
f(x) & = & 5x^2 - \frac{1}{x}+ 3\\
g(x) & = & x^2-k
\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$?### Response: I need to find $k$ such that $f(2) - g(2) = 2$.
To do that, I will plug in $x = 2$ into both functions and simplify.
For $f(2)$, I get $f(2) = 5(2)^2 - \frac{1}{2} + 3 = 20 - \frac{1}{2} + 3 = 22.5$.
For $g(2)$, I get $g(2) = (2)^2 - k = 4 - k$.
Now I can set up an equation: $f(2) - g(2) = 2$.
Substituting the values, I get $22.5 - (4 - k) = 2$.
Simplifying, I get $18.5 + k = 2$, so $k = -16.5$.
I can check my answer by plugging it back into the original equation and seeing if it works.
If $k = -16.5$, then $g(2) = 4 - (-16.5) = 20.5$.
And then $f(2) - g(2) = 22.5 - 20.5 = 2$, which is correct.
So the value of $k$ is -16.5.