### 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: To find the value of $k$, I need to substitute $x=2$ into both functions and set them equal to each other with a difference of 2.
That means I need to solve the equation $$f(2) - g(2) = 2$$
Plugging in $x=2$ into $f(x)$, I get $$f(2) = 5(2)^2 - \frac{1}{2} + 3$$
Simplifying, I get $$f(2) = 20 - \frac{1}{2} + 3 = \frac{45}{2}$$
Plugging in $x=2$ into $g(x)$, I get $$g(2) = (2)^2 - k$$
Simplifying, I get $$g(2) = 4 - k$$
Now I can substitute these values into the equation and get $$\frac{45}{2} - (4 - k) = 2$$
I can solve for $k$ by isolating it on one side of the equation.
First, I need to get rid of the fractions by multiplying everything by 2.
This gives me $$45 - 2(4 - k) = 4$$
Expanding the parentheses, I get $$45 - 8 + 2k = 4$$
Simplifying, I get $$2k + 37 = 4$$
Subtracting 37 from both sides, I get $$2k = -33$$