Topic 14 Rational Functions

14.1 The Law of Lever

“Give me a fulcrum and a place to stand, I will move the world.” by Archimedes of Syracuse

In volume I of his book “On the Equilibrium of Planes”, Archimedes proved that magnitudes are in equilibrium at distances reciprocally proportional to their weights. See the video Law of the Lever on youtube for an animated explanation.

Suppose there is a infinite long lever with a load 100 newton that is placed 1 meters away the fulcrum, the pivoting point.

  • Can you find the force needed to balance the load in terms of the distance away from the fulcrum?
  • How much force will be needed if it is place 5 meters away from the fulcrum?

14.2 The Domain of a Rational Function

A rational function \(f\) is defined by an equation \(f(x)=\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and the degree of \(q(x)\) is at least one. Since the denominator cannot be zero, the domain of \(f\) consists all real numbers except the numbers such that \(q(x)=0\)

Example 14.1 Find the domain of the function \(f(x)=\frac{1}{x-1}\).


Solve the equation \(x-1=0\), we get \(x=1\). Then the domain is \(\{x\mid x\neq 1\}\). In interval notation, the domain is \[ (-\infty, 1)\cup (1,\infty). \]

Example 14.2 Suppose that the cost in dollars to produce x toy cars is given by C(x)=5,000+5x. How many toy cars would the factory have to produce in order for the average production cost $10 per item?

Solution. The average cost is defined as \[A(x)=\frac{C(x)}{x}.\] The number of toy cars should be produced to get the average cost $10 is the solution of the equation \(A(x)=10\), that is \[ \begin{aligned} \frac{5000+5x}{x}&=10\\ 5000+5x&=10x\\ 5000&=5x\\ x&=1000. \end{aligned} \]

The factory should produce 1000 toy cars for the average cost $10 per item.

14.3 Practice

Problem 14.1 Find the domain of each function. Write in interval notation.

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