Natural Logarithm

Question

These are the questions:

Consider any function of the form f(x)= ln u(x)

(i) if we wish to find the domain of our function, what must be true of our u(x)? Explain in a complete sentence.

(ii) Now suppose u(x) =x^3-12x. Write the function f(x) now, and then find its domain, using (i). Give exact answer (not decimal approximations). Hint: you might start with factoring u(x), setting it equal to zero, and then making a signs diagram for u(x). Write a complete sentence answer to the domain question, and give your answer in interval notation.

Answer 1

0 < x(x² - 12)

Both factors are positive:

x > 0 and x² - 12 > 0

x > 0 and x² > 12

x > 0 and x > ±2√3

Since x > 0 (only posirive), x > 2√3

Or

Both factors are negative

x < 0 and x² < 12

x < 0 and x < ±2√3

Since x < 0 (only negative), x < -2√3

(-∞, -2√3) U (2√3, ∞)