Operations with Rational Functions

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What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \neq 0 \).

Front

What does it mean to simplify a rational function?

Back

Simplifying a rational function involves reducing it to its simplest form by canceling common factors in the numerator and denominator.

Front

What is the domain of a rational function?

Back

The domain of a rational function is all real numbers except where the denominator is zero.

Front

How do you find the vertical asymptotes of a rational function?

Back

Vertical asymptotes occur at the values of \( x \) that make the denominator zero, provided these values do not also make the numerator zero.

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What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as \( x \) approaches infinity or negative infinity.

Front

How do you perform operations (addition, subtraction) on rational functions?

Back

To add or subtract rational functions, find a common denominator, rewrite each function with that denominator, and then combine the numerators.

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What is the process for multiplying rational functions?

Back

To multiply rational functions, multiply the numerators together and the denominators together, then simplify if possible.

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What is the process for dividing rational functions?

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To divide rational functions, multiply by the reciprocal of the divisor, then simplify.

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How do you solve a rational equation?

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To solve a rational equation, find a common denominator, eliminate the denominators, and solve the resulting polynomial equation.

Front

What is a hole in the graph of a rational function?

Back

A hole occurs in the graph of a rational function at values of \( x \) that make both the numerator and denominator zero.