Find all values of x for which the graph of f lies above the graph of g. (Enter your answer using interval notation.) f(x) = 36x; g(x) = 1/x

Question

the answer is not (-infinity,-1/6)u(1/6,infinity)

Answer 1

We want to determine the values of x so that f(x) - g(x) > 0, i.e. f(x) > g(x).

Let's call C(x) = f(x) - g(x) = 36x - 1/x

When does C(x) = 0?

36x - 1/x = 0

36x² - 1 = 0

x² = 1/36

x = ±1/6

Note that x cannot equal 0.

Picture a number line displaying ---- -1/6 ---- (0) ----- 1/6 -------------

To the left of -1/6 pick a test number, say -1. C(-1) = -36 +1 which is negative. Numbers in this interval do not make f(x) - g(x) > 0.

Try a number between -1/6 and 0, say -1/12.

C(-1/12) = -3 + 12 which is positive. So numbers in (-1/6, 0) do make f(x) - g(x) > 0.

Now between 0 and 1/6, choose 1/12.

C(1/12) = 3 - 12. Negative.

Finally to the right of 1/6, choose 1.

C(1) = 36 - 1. Positive. Numbers in the interval (1/6, ∞) make f(x) - g(x) > 0.

Final solution set in interval notation: (-1/6, 0) U (1/6, ∞).

Check the link to a Desmos graph shown in the comments section for confirmation.