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College Algebra: Graphs, Functions, and Systems

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이 집합의 용어 (19)

  • What is the definition of a function in algebra?
    A function is a relation where each input has exactly one output.
  • What is the vertical line test used for?
    The vertical line test determines if a graph represents a function by checking if any vertical line crosses the graph more than once.
  • How do you find the domain of a function from its graph?
    The domain is the set of all x-values covered by the graph.
  • What is the range of a function?
    The range is the set of all possible output values (y-values) of the function.
  • What does it mean to compose two functions?
    Function composition means applying one function to the result of another, written as \((f \circ g)(x)\).
  • How do you perform addition of two functions \(f(x) + g(x)\)?
    Add the outputs of the functions for each input: \((f+g)(x) = f(x) + g(x)\).
  • What is the difference between the domain of \(f(x)\\) and \(g(x)\\) and the domain of their sum?
    The domain of \(f+g\) is the intersection of the domains of \(f\) and \(g\).
  • What is a system of linear equations?
    A system of linear equations is a set of two or more linear equations with the same variables.
  • What are the common methods to solve systems of linear equations?
    Common methods include graphing, substitution, and elimination.
  • What does the solution of a system of linear equations represent?
    The solution is the point(s) where the graphs of the equations intersect.
  • What is a system of inequalities?
    A system of inequalities is a set of two or more inequalities with the same variables.
  • How do you graph a system of inequalities?
    Graph each inequality and find the overlapping shaded region that satisfies all inequalities.
  • What does the solution region of a system of inequalities represent?
    It represents all points that satisfy every inequality in the system.
  • What is the purpose of graphing techniques in algebra?
    Graphing techniques help visualize functions and relations to analyze behavior and solutions.
  • How do you find the inverse of a function graphically?
    The inverse is the reflection of the function's graph across the line \(y=x\).
  • What is the effect of shifting a graph vertically?
    Adding a constant to the function shifts the graph up or down without changing its shape.
  • What is the effect of shifting a graph horizontally?
    Replacing \(x\) with \(x-h\) shifts the graph right by \(h\) units.
  • What is the difference between function operations and composition?
    Function operations combine outputs (add, subtract, multiply, divide), while composition applies one function inside another.
  • How do you determine if two linear equations are dependent, independent, or inconsistent?
    Dependent: same line; Independent: intersect at one point; Inconsistent: parallel lines with no intersection.