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Precalculus: Domain, Range, and Difference Quotient

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Termini in questo insieme (20)

  • What is the domain of a function?
    The domain is the set of all possible input values (first coordinates) for which the function is defined.
  • What is the range of a function?
    The range is the set of all possible output values (second coordinates) that the function can produce.
  • How do you find the domain of a function defined by an equation?
    Start with all real numbers, then exclude values that make denominators zero or radicands negative for even roots.
  • What is interval notation for domain or range?
    Interval notation expresses the domain or range as intervals, e.g., [−5, 3] means all values from −5 to 3 inclusive.
  • What is set-builder notation for domain or range?
    Set-builder notation describes the set using a property, e.g., {x | −5 ≤ x ≤ 3} means all x such that x is between −5 and 3.
  • Given a graph with x-values from −5 to 3, how is the domain expressed?
    Domain in interval notation: [−5, 3]; in set-builder notation: {x | −5 ≤ x ≤ 3}.
  • Given a graph with y-values from −1 to 4, how is the range expressed?
    Range in interval notation: [−1, 4]; in set-builder notation: {y | −1 ≤ y ≤ 4}.
  • What is the difference quotient formula?
    The difference quotient is \(\frac{f(x+h)-f(x)}{h}\), where h ≠ 0.
  • Find the difference quotient for \(f(x)=5\).
    Difference quotient = 0, since \(\frac{5-5}{h}=0\).
  • Find the difference quotient for \(f(x)=2x+5\).
    Difference quotient = 2, since \(\frac{2(x+h)+5-(2x+5)}{h} = 2\).
  • Find the difference quotient for \(f(x)=5x+1\).
    Difference quotient = 5, since \(\frac{5(x+h)+1-(5x+1)}{h} = 5\).
  • Find the difference quotient for \(f(x)=2x^2+5\).
    Difference quotient = \(4x+2h\) after simplification.
  • How to find the domain of \(f(x)=\frac{3}{2x-5}\)?
    Exclude values making denominator zero: domain is all real numbers except \(x=\frac{5}{2}\).
  • How to find the domain of \(g(x)=\sqrt{4+3x}\)?
    Set radicand ≥ 0: \(4+3x \geq 0\), so domain is \(x \geq -\frac{4}{3}\).
  • What is the domain of the function \(f(x)=3x+2\)?
    Domain is all real numbers, since it is a linear function with no restrictions.
  • How to express the domain of a function graphically?
    Identify all x-values covered by the graph and write them in interval or set-builder notation.
  • How to express the range of a function graphically?
    Identify all y-values covered by the graph and write them in interval or set-builder notation.
  • What does it mean if a function has an absolute maximum or minimum?
    The function reaches its highest or lowest value over its entire domain.
  • What is a piecewise-defined function?
    A function defined by different expressions over different parts of its domain.
  • What is the average rate of change of a function?
    The change in output divided by the change in input over an interval, similar to slope.