Problem 76
Question
What is a rational function?
Step-by-Step Solution
Verified Answer
A rational function is a function that can be defined by a ratio of two polynomials. For instance, \(f(x) = \frac{x^2 + 3x + 2}{x - 1}\) is a rational function.
1Step 1: Define Rational Function
A rational function is any function that can be defined by a rational fraction, i.e., an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers, they could be taken in any field K. In this context, a field K is a set equipped with two operations, called addition and multiplication.
2Step 2: Explain division by zero
A rational function is not defined when the denominator is zero. When the denominator of a fraction is 0, the fraction represents division by zero, and the value of the fraction is undefined.
3Step 3: Provide example of rational function
Here's an example of a rational function: \(f(x) = \frac{x^2 + 3x + 2}{x - 1}\) . Note, it's in the form of a ratio of two polynomials, and is not defined when \(x = 1\) because that would make the denominator zero.
Other exercises in this chapter
Problem 75
In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x
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In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{4}+8 x^{
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In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{5}+5 x^{
View solution Problem 78
In Exercises \(78-79,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \(200 \mathrm{M} \text { OUT }]\) feature to
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