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Introduction to Rational Functions quiz

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  • What is a rational function?
    A rational function is a function formed by dividing one polynomial by another, written as p(x)/q(x), where both p(x) and q(x) are polynomials.
  • Why can't the denominator of a rational function be zero?
    The denominator cannot be zero because division by zero is undefined, which restricts the domain of the function.
  • How do you find the domain of a rational function?
    Set the denominator equal to zero and solve for x; the solutions are the values excluded from the domain.
  • What is the domain of the function 3/(3x-12)?
    The domain is all real numbers except x = -4, because 3x-12 equals zero when x = -4.
  • For f(x) = (x+5)/(x^2-25), which x-values are excluded from the domain?
    x = 5 and x = -5 are excluded because they make the denominator zero.
  • What is the first step in simplifying a rational function to lowest terms?
    The first step is to factor both the numerator and the denominator.
  • Why should you determine the domain before simplifying a rational function?
    You should determine the domain first to avoid missing any restrictions that might be lost when simplifying.
  • How do you simplify the function 3/(3x+12)?
    Factor the denominator to get 3(x+4), then cancel the common factor of 3, leaving 1/(x+4).
  • What is the simplified form of f(x) = (x+5)/(x^2-25)?
    It simplifies to 1/(x-5) after factoring and canceling the common factor x+5.
  • If you simplify a rational function before finding the domain, what might happen?
    You might miss some domain restrictions because common factors could be canceled out.
  • What does it mean to write a rational function in lowest terms?
    It means to factor and cancel any common factors in the numerator and denominator.
  • What is a domain restriction in the context of rational functions?
    A domain restriction is a value of x that makes the denominator zero and is therefore excluded from the domain.
  • How do you write the domain of 1/(x-1) in set notation?
    The domain is {x | x ≠ 1}, meaning all real numbers except x = 1.
  • What is the difference of squares factoring used for in rational functions?
    It is used to factor denominators like x^2-25 into (x+5)(x-5) to help simplify the function.
  • Can a rational function have more than one domain restriction? Give an example.
    Yes, for example, f(x) = (x+5)/(x^2-25) has domain restrictions at x = 5 and x = -5.