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

(i)

D = {x | u(x) > 0}. Only numbers greater than 0 have a natural logrithm. Therefore the domain of u(x) is only the values of x that result in u(x) being positive.

(ii)

u(x) = x³ - 12x

0 < x³ - 12x

0 < x(x² - 12)

The factors are either both positive or both negative. Can you deteremine the values of x that make it so?