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Continuous Functions - Problem 2

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Another property of continuous functions is that the product and quotient of f(x) and g(x) are continuous. This means that if f(x) and g(x) are continuous, then f(x)*g(x) is continuous. Also, f(x)/g(x) is continuous. Since polynomials are continuous, the product of polynomials and the quotient of polynomials are also continuous. Recall that a function is continuous if it is continuous at all points at which the function is defined.

Let's talk about another rule for continuous functions. If two functions f, and g are continuous functions, then the product of f, and g, and the quotient of f over g are both continuous functions wherever they're defined.

Now let's do an example. Explain why r(x) equals 24x over x² minus 9 is a continuous function. Well, first of all 24x, and x² minus 9 are both polynomial functions, so they are continuous. So 24x, and x² minus 9 are continuous, because they are polynomials.

Now what I have here is a quotient of two continuous functions just like in rule 4 here. So by rule 4, 24x over x² minus 9 is continuous wherever it's defined. That means it's continuous provided that x² minus 9 is not 0. This is going to defined as long as the denominator is 0. So x² minus 9 is not 0.

Now that means that x is not equal to plus or -3. What we have here, because this is our function r(x). r(x) is continuous for all real numbers except plus or -3. Now r(x) is an example of rational functions. Remember that rational functions are functions that are made by taking one polynomial divided by another polynomial. All rational functions are continuous functions which means they're continuous wherever they're defined.

Let's take a look. So we just showed that r(x) this function is continuous wherever it's defined. It's defined for all x not equal to plus or -3. This function s(x) is continuous wherever it's defined. Remember that rational functions are only undefined when they're denominators are 0. This denominator can never equal 0. That means this is continuous for all real numbers.

Finally, the last function 5x² plus 9x minus 2 over x plus 2, this is continuous wherever it's defined. It's only undefined when x equals -2. So this is continuous for x not equal to -2. All rational functions are continuous wherever they're defined. They're only undefined when their denominators are 0.

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