If f and g are functions, where f(x) = x^3 -10x^2 +27x – 18 and g(x) = x^3 – x^2 – 6x, which of the following gives a relationship between f and g?

A) g(x) = 3 f(x)
B) g(x) = f(x) – 3
C) g(x) = f(x) + 3
D) g(x) = f(x – 3)
E) g(x) = f(x + 3)

Can you solve w/o graphing?

Yes!

Note that the first term in both functions is x^3, so the relationship isn’t multiplication. Eliminate choice A.

Note also that it’s obvious that g(x) isn’t just 3 bigger or 3 smaller than f(x), so you can also eliminate choices B and C without any real work.

From there, it gets a little tricky. Focus on the constant terms, though. Note that f(x) has a –18 in it that goes away in g(x). Note also that when we plug x-3 or x+3 in for x, we’ll have a bunch of binomials to expand, but the only way that –18 goes away is if those binomial expansions spit out a positive 18. Without doing all the math, you should be confident that choice E is the one that will do that. To test it quickly, though, set x=0 (since this does need to be true when x=0).

    \begin{align*}g(0)&=f(0+3)\\g(0)&=f(3)\\0^3 - 0^2 - 6(0)&=3^3 -10(3)^2 +27(3) - 18\\0&=27-90+81-18\\0&=0\end{align*}

Yep! That worked nicely.