Finding Domain & Range in Functions
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What is the domain of the function f?

  • All real numbers less than −1
  • All real numbers except 2
  • All real numbers
  • All real numbers greater than −2/3 (correct)
  • What is the range of the function f defined in the content?

  • y | y ≥ −1 (correct)
  • y | y = 0
  • y | y ≠ 0
  • y | y < −1
  • What is the output of the function f(x) = 3 - x when x = 2?

  • 3
  • 1 (correct)
  • 2
  • 4
  • For the function f(x) = 3/(x − 2), which statement about its domain is correct?

    <p>The domain is {x | x ≠ 2}</p> Signup and view all the answers

    Which statement correctly describes the domain of the function f(x) = x^2?

    <p>The domain consists of all real numbers.</p> Signup and view all the answers

    Using set-builder notation, how would you express the domain of the function f(x) = x for nonnegative real numbers?

    <p>{x | x ≥ 0}</p> Signup and view all the answers

    What is the resulting equation when solving for x in f(x) = 3?

    <p>x = 3y + 2</p> Signup and view all the answers

    What is the interval notation for the set of real numbers greater than 1 but less than 5?

    <p>(1, 5)</p> Signup and view all the answers

    Which of the following correctly identifies the range of f(x) = 3/(x − 2)?

    <p>y | y ≠ 0</p> Signup and view all the answers

    When modeling real-world applications, which of the following tools can be used to represent a function?

    <p>A formula, a table, or a graph</p> Signup and view all the answers

    Which function restricts its output to real numbers when the input is nonnegative?

    <p>f(x) = sqrt(x)</p> Signup and view all the answers

    In the temperature function example provided, which variable is the input variable?

    <p>Time after midnight</p> Signup and view all the answers

    What is the output set when applying the function f(x) = 3 - x to the domain D = {1, 2, 3}?

    <p>{3, 2, 1}</p> Signup and view all the answers

    What type of function is associated with the domain of all real numbers?

    <p>Quadratic function</p> Signup and view all the answers

    Which statement is true regarding the function f(x) = 4 - 2x + 5?

    <p>Its domain is all real numbers.</p> Signup and view all the answers

    How can you denote the set of all real numbers between two values a and b, inclusive of a but exclusive of b?

    <p>[a, b)</p> Signup and view all the answers

    What characterizes an even function?

    <p>f(-x) = f(x)</p> Signup and view all the answers

    Which of the following functions is classified as odd?

    <p>f(x) = 3x/(x^2 + 1)</p> Signup and view all the answers

    Which of the following descriptions is true about odd functions?

    <p>They are symmetric about the origin.</p> Signup and view all the answers

    If f(x) = 2x^5 - 4x + 5, what can be concluded about its symmetry?

    <p>The function is neither even nor odd.</p> Signup and view all the answers

    Which of the following statements is NOT true regarding the absolute value function?

    <p>It makes all outputs positive.</p> Signup and view all the answers

    How is the absolute value function defined for positive inputs?

    <p>|x| = x</p> Signup and view all the answers

    For a function to be considered even, which of the following must hold true?

    <p>f(-x) = f(x)</p> Signup and view all the answers

    What is the general form of a function that is neither even nor odd?

    <p>Contains both even and odd-powered terms with no symmetry.</p> Signup and view all the answers

    What is the value of f(5) for the function defined by f(x) = 3x + 1 for x ≥ 2 and f(x) = x^2 for x < 2?

    <p>16</p> Signup and view all the answers

    Evaluate f(a + h) for the function f(x) = 3x^2 + 2x - 1.

    <p>3a^2 + 6ah + 3h^2 + 2a + 2h - 1</p> Signup and view all the answers

    What is the domain of the function f(x) = (x - 4)^2 + 5?

    <p>All real numbers</p> Signup and view all the answers

    What is the minimum value of the function f(x) = (x - 4)^2 + 5?

    <p>5</p> Signup and view all the answers

    For the function f(x) = 3/(x - 2), what is its domain?

    <p>x ≠ 2</p> Signup and view all the answers

    What is the range of the function f(x) = (x - 4)^2 + 5?

    <p>All real numbers greater than or equal to 5</p> Signup and view all the answers

    What is the value of f(-2) for the function f(x) = 3x^2 + 2x - 1?

    <p>7</p> Signup and view all the answers

    For the piecewise function f(x) defined as f(x) = 3x + 1 for x ≥ 2, what is f(1)?

    <p>Undefined</p> Signup and view all the answers

    Which of the following represents a function of the form $f(x) = x^2 + 1$?

    <p>The set of points (−3, 10) and (3, 10)</p> Signup and view all the answers

    What is the y-intercept of the function $f(x) = -x^2$?

    <p>0</p> Signup and view all the answers

    Which of the following statements is true regarding the function $f(x) = x^3$?

    <p>It is an odd function.</p> Signup and view all the answers

    What is the domain of the function $f(x) = x^3$?

    <p>All real numbers</p> Signup and view all the answers

    What can be concluded from the vertical line test for the graph of a function?

    <p>If a vertical line crosses the graph more than once, it does not represent a function.</p> Signup and view all the answers

    For the function $f(x) = -x^2$, what is the range?

    <p>All numbers less than or equal to 0</p> Signup and view all the answers

    For which intervals is the function $f(x) = x^2 + 1$ increasing?

    <p>From 0 to $ ext{∞}$</p> Signup and view all the answers

    Which point is an x-intercept for the function $f(x) = x^3$?

    <p>(0, 0)</p> Signup and view all the answers

    Study Notes

    Finding Domain & Range

    • Domain is the set of all possible input values (x-values) for which a function is defined.
    • Range is the set of all possible output values (y-values) that the function can produce.
    • To find the domain, consider what values of x would make the function undefined (e.g., division by zero).
    • To find the range, consider what values of y can be achieved by the function.

    Domain and Range Examples:

    • f(x) = (x-4)^2 + 5:
      • Domain: All real numbers (represented as (-∞, ∞))
      • Range: y ≥ 5 (represented as {y | y ≥ 5})
    • f(x) = 3x + 2 - 1:
      • Domain: All real numbers
      • Range: All real numbers
    • f(x) = 3 / (x-2):
      • Domain: All real numbers except x = 2 (represented as {x | x ≠ 2})
      • Range: All real numbers except y = 0 (represented as {y | y ≠ 0})

    Representing Functions:

    • Functions can be represented using tables, graphs, and formulas.
    • Tables: Organize input (x) and output (y) values.
    • Graphs: Visualize the function's relationship between input and output, plotting (x, y) points.
    • Formulas: Express the function's rule, defining output (y) in terms of input (x).

    Natural Domain:

    • The domain of a function is considered the natural domain if no specific domain is given.
    • This often means all real numbers where the function is defined.
    • The square root function f(x) = √x has a natural domain of non-negative real numbers, as the square root of a negative number is not a real number.

    Set-builder Notation:

    • Defines a set of numbers using a specific property or condition.
    • Example: {x | 1 < x < 5} represents the set of all real numbers x where x is greater than 1 and less than 5.
    • The notation {x | x has some property} is read as "the set of real numbers x such that x has some property".

    Interval Notation:

    • Represents a range of numbers.
    • Example: (1, 5) represents the interval of all numbers greater than 1 and less than 5 (excluding 1 and 5).

    Even and Odd Functions:

    • Even Function: f(x) = f(-x) for all x in the domain of f.
    • Odd Function: f(-x) = -f(x) for all x in the domain of f.
    • Even Functions: Symmetric about the y-axis.
    • Odd Functions: Symmetric about the origin (rotational symmetry).

    Absolute Value Function:

    • Defined as f(x) = -x for x < 0 and f(x) = x for x ≥ 0.
    • Represented as |x|.
    • Makes all nonzero inputs positive, but outputs 0 if x = 0.
    • Range: y ≥ 0 (represented as {y | y ≥ 0}).
    • Even function.

    Vertical Line Test:

    • Determines if a graph represents a function.
    • If a vertical line intersects the graph at more than one point, it is not a function.
    • If a vertical line intersects the graph at only one point, it is a function.

    Function Combinations:

    • Sum (f+g): (f+g)(x) = f(x) + g(x)
    • Difference (f-g): (f-g)(x) = f(x) - g(x)
    • Product (f⋅g): (f⋅g)(x) = f(x)⋅g(x)
    • Quotient (f/g): (f/g)(x) = f(x) / g(x) (where g(x) ≠ 0)
    • Domain of Combinations: The domain of the new function is the intersection of the domains of f and g, excluding any values that would make the denominator zero in the case of the quotient (f/g).

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    Description

    This quiz focuses on identifying the domain and range of various functions. You will learn to determine the set of possible input and output values for different types of functions and scenarios. Test your understanding with examples and conceptual challenges.

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