### 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 the value of $k$ that makes the difference of the two functions equal to 2 when $x=2$.
To do that, I can plug in $x=2$ into both functions and then set their difference equal to 2.
For $f(2)$, I get $$f(2) = 5(2^2) - \frac{1}{2} + 3 = 20 - \frac{1}{2} + 3 = \frac{45}{2}$$
For $g(2)$, I get $$g(2) = (2^2) - k = 4 - k$$
Now I can set their difference equal to 2 and solve for $k$: $$f(2) - g(2) = 2$$ $$\frac{45}{2} - (4 - k) = 2$$
I can multiply both sides by 2 to get rid of the fractions: $$45 - 2(4 - k) = 4$$ $$45 - 8 + 2k = 4$$
I can simplify and isolate $k$: $$2k = 4 - 45 + 8$$ $$2k = -33$$ $$k = -\frac{33}{2}$$