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Rational functions: FAQ

What's the point of reducing a rational expression to lowest terms?

Reducing a rational expression to lowest terms makes it simpler to work with. It can make it easier to see common factors, and it can help us more easily compare two rational expressions to each other.

What is "end behavior" when it comes to rational functions?

End behavior refers to the way a function behaves as its input approaches infinity (either positively or negatively). For rational functions, we often look at the horizontal asymptote, or the line the function gets closer and closer to as we move further and further away from the origin on the x-axis.

Why do rational functions sometimes have discontinuities?

A rational function will have a discontinuity at any x-value where the denominator is equal to 0. This is because division by zero is undefined, so the function doesn't produce an output for that input.

How do I graph a rational function?

There are a few steps we can take to graph a rational function. We can start by finding the x- and y-intercepts, and identifying any vertical or horizontal asymptotes. We can also look for symmetry or plot a few additional points in order to get a good idea of the shape of the graph.

How can we use rational functions to model real-world situations?

Rational functions can be used in a variety of ways to model real-world situations. For example, we can use a rational function to model the speed of a car that's braking, or the amount of a drug in a person's system over time.

How do we multiply rational expressions?

We can use the same rules of fractions that we learned in earlier math classes.
To multiply two rational expressions, we multiply the numerators (the top parts of the fractions) together, and we multiply the denominators (the bottom parts of the fractions) together.
We can also cancel common factors when multiplying rational expressions, following these steps:
  1. Factor the numerators and denominators of the rational expressions so that you can identify any common factors.
  2. Identify any factors that are present in both the numerator and denominator of one or more of the rational expressions.
  3. Cancel the common factors by crossing them out or rewriting the rational expression without the common factors.

How do we divide rational expressions?

We can use the same rules of fractions that we learned in earlier math classes.
To divide one rational expression by another, we write the two expressions out with the division symbol between them. Flip (or invert) the fraction on the right side of the division symbol, so that the numerator and denominator switch places. Change the division symbol to a multiplication symbol, and multiply the two expressions together.

How do we add and subtract rational expressions?

To add or subtract rational expressions, you need to find a common denominator, just like when adding or subtracting fractions.
  1. Factor the denominators of each rational expression, if they are not already factored.
  2. Determine the least common denominator (LCD) by finding the smallest multiple that each denominator will divide evenly into.
  3. Use the LCD to rewrite each rational expression so that they all have the same denominator.
  4. Add or subtract the numerators as indicated, and keep the denominator the same.

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