Topic 15 Radical Functions

15.1 Speed of a Tsunami

A tsunami is generally referred to is a series of waves on the ocean caused by earthquakes or other events that cause sudden displacements of large volumes of water. In ideal situation, the velocity v of a wave at where the water depth is d meters is approximately v=\sqrt{9.8d}. The wave will slow down when closer to the coast but will be higher.

Suppose a tsunami was caused by earthquake somewhere 10000 meters away the coast of California. The depth of the water where the tsunami was generated is 5000 meter.

  • What’s the initial speed of the tsunami?
  • What’s the speed of the tsunami at where the water depth is 2000.
  • Suppose the speed wouldn’t decrease, how long it takes the tsunami reach the coast?

15.2 The Domain of a Radical Function

A radical function f is defined by an equation f(x)=\sqrt[n]{r(x)}, where r(x) is an algebraic expression. For example f(x)=\sqrt{x+1}. When n is odd number, r(x) can be any real number. When n is even, r(x) has to be nonnegative, that is r(x)\geq 0 so that f(x) is a real number.

Example 15.1 Find the domain of the function f(x)=\sqrt{x+1}.

Solution.

Since the index is 2 which is even, the function has real outputs only if the radicand x+1\geq 0. Solve the inequality, we get x\geq -1. In interval notation, the domain is [-1,\infty).

Example 15.2 For a pendulum clock, the period T of the pendulum is approximately modeled by the following function of the length L of the pendulum: T=2\sqrt{L} where L and T are measured in meters and seconds respectively.

  • If the length of the pendulum is 4 meters, what is the period?
  • If the period of a pendulum clock is 1 second, how long should be the pendulum?
  • Solution.
  • Because the length of the pendulum is 4 meters, that is L=4. Then the period is T=2\cdot\sqrt{4}=4 seconds.
  • Because the period of the pendulum is 1 second, that is T=1. Then the length L is the solution of the equations 2\sqrt{L}=1. \begin{aligned} 2\sqrt{L}&=1\\ \sqrt{L}&=\frac12\\ L&=\left(\frac12\right)^2\\ L&=\frac14. \end{aligned} So the length of the pendulum should be 1/4 in so that the period is 1 second.

15.3 Practice

Problem 15.1 Find the domain of each function. Write in interval notation.

  1. f(x)=1-\frac{2x}{x+3}
  2. f(x)=\frac{x-2}{x^2-4}
  3. f(x)=\sqrt{1-x^2}
  4. f(x)=-\sqrt{\frac{1}{x-5}}