we need to find an x so that f(x) = c, for any c we choose. so x should be "some expression (formula) in c".
what is f(x)? by the definition:
$\displaystyle f(x) = 2x^3 - 4$. so if f(x) = c, we have:
$\displaystyle 2x^3 - 4 = c$. now, we need to try to "solve for x".
$\displaystyle 2x^3 = c+4$ (adding 4 to both sides)
$\displaystyle x^3 = \frac{c+4}{2}$ (dividing both sides by 2)
$\displaystyle x = \sqrt[3]{\frac{c+4}{2}}$ (taking the cube root of both sides). this is the x we are looking for (we hope).
to check, we verify that f(x) is indeed c, when $\displaystyle x = \sqrt[3]{\frac{c+4}{2}}$.
$\displaystyle f\left(\sqrt[3]{\frac{c+4}{2}}\right) = 2\left(\sqrt[3]{\frac{c+4}{2}}\right)^3 - 4$
$\displaystyle = 2\left(\frac{c+4}{2}\right) - 4 = (c+4) - 4 = c + (4-4) = c+0 = c$,
so we indeed found an x that f sends to c.