CH 2: Functions and Relations
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Questions and Answers

What is a function in mathematical analysis?

  • A relation between two sets where each element in the domain is associated with more than one element in the range
  • A relation between two sets where each element in the domain is associated with exactly one element in the range (correct)
  • A relation between two sets where each element in the domain has no association with any element in the range
  • A relation between two sets where each element in the domain is associated with multiple elements in the range

    • How is a function denoted when mapping every element from set A to set B?

  • $x = f(y)$
  • $y = f(x)$ (correct)
  • $f(x) = y$
  • $y = f(1/x)$

    • What conditions must a function satisfy for its inverse to exist?

  • The function must be injective and surjective (correct)
  • The function must be injective but not necessarily surjective
  • The function must be surjective but not necessarily injective
  • The function must not be bijective

    • What does an inverse function do to the mapping of the original function?

    <p>It reverses the assignment made by the original function</p> Signup and view all the answers

    What form does a linear function take?

    <p>$y = mx + c$</p> Signup and view all the answers

    Which type of function mapping is required for an inverse to exist?

    <p>One-to-one (bijective)</p> Signup and view all the answers

    What does every element of the domain map to in a surjective function?

    <p>Exactly one element in the range</p> Signup and view all the answers

    In a linear function, what does 'm' represent?

    <p>The slope</p> Signup and view all the answers

    'Every element of the domain maps to a unique element of the range' describes which property of a function?

    <p><em>Injective</em></p> Signup and view all the answers

    What is necessary for a linear function to have an inverse?

    <p>Being both injective and surjective</p> Signup and view all the answers

    What is the slope of the linear function $f(x) = 2x + 3$?

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

    What is the y-intercept of the linear function $f(x) = 2x + 3$?

    <p>3</p> Signup and view all the answers

    What is the formula for the inverse of the linear function $f(x) = 2x + 3$?

    <p>$f^{-1}(x) = rac{x - 3}{2}$</p> Signup and view all the answers

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

    <p>3</p> Signup and view all the answers

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

    <p>$f^{-1}(x) = ext{log}_3(x)$</p> Signup and view all the answers

    What is the coefficient $a$ in the quadratic function $f(x) = 2x^2$?

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

    What restriction is needed to ensure the inverse of a quadratic function is also a function?

    <p>The domain must be restricted to $x \geq 0$.</p> Signup and view all the answers

    What is the key concept that determines if a graph represents a function?

    <p>Vertical Line Test</p> Signup and view all the answers

    What is the key concept that determines if a function has an inverse function?

    <p>Horizontal Line Test</p> Signup and view all the answers

    Which type of function is represented by a straight line?

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

    What is the essential characteristic that a function must possess to have an inverse?

    <p>The function must be one-to-one (injective).</p> Signup and view all the answers

    What is the graphical representation of a linear function with a non-zero slope?

    <p>A straight line</p> Signup and view all the answers

    If $f(x) = 4x^2$, what restriction must be placed on the domain to ensure the inverse function exists?

    <p>$x \geq 0$</p> Signup and view all the answers

    What is the inverse function of $f(x) = 2^x$?

    <p>$f^{-1}(x) = \log_2(x)$</p> Signup and view all the answers

    Which test determines if a function has an inverse?

    <p>The horizontal line test</p> Signup and view all the answers

    What does the notation $f^{-1}(x)$ represent?

    <p>The value of the inverse function at $x$</p> Signup and view all the answers

    If $f(x) = 3x - 1$, what is the formula for the inverse function?

    <p>$f^{-1}(x) = \frac{x - 1}{3}$</p> Signup and view all the answers

    Which type of function is represented by a rapidly increasing or decreasing curve that does not touch the x-axis?

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

    What is the key concept used to determine if a graph represents a function?

    <p>The vertical line test</p> Signup and view all the answers

    If $f(x) = x^2$, what is the range of the function?

    <p>Non-negative real numbers</p> Signup and view all the answers

    If $f(x) = 2x + 3$, what is the value of $f^{-1}(7)$?

    <p>4</p> Signup and view all the answers

    If $f(x) = x^2$ and $g(x) = \sqrt{x}$, which of the following is true?

    <p>$f(x)$ is the inverse of $g(x)$</p> Signup and view all the answers

    If $f(x) = 3^x$, what is $f^{-1}(81)$?

    <p>4</p> Signup and view all the answers

    Which condition must be satisfied for a function $f(x)$ to have an inverse?

    <p>$f(x)$ must be one-to-one and onto</p> Signup and view all the answers

    If $g(x) = \frac{1}{x}$, which of the following represents $g^{-1}(x)$?

    <p>$x$</p> Signup and view all the answers

    Which of the following functions is not a one-to-one function?

    <p>$k(x) = x^2 - 2</p> Signup and view all the answers

    If $f(x) = 2x - 3$ and $g(x) = \frac{x + 3}{2}$, what is $g(f(x))$?

    <p>$x$</p> Signup and view all the answers

    If $f(x) = \frac{1}{x+2}$, what is the domain of $f(x)$?

    <p>All real numbers except $x = -2</p> Signup and view all the answers

    For a function to have an inverse, it must be both:

    <p>Injective and surjective</p> Signup and view all the answers

    What is the condition for a function to be considered bijective?

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

    In a linear function, what does 'c' represent?

    <p>Y-intercept</p> Signup and view all the answers

    'Every element of the range is mapped from the domain' describes which property of a function?

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

    What type of function is represented by a horizontal line on a graph?

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

    'One-to-one' is synonymous with which property of a function?

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

    'Onto' is another term used to describe which property of a function?

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

    'Bijective' functions are composed of which two key properties?

    <p>Injective and surjective</p> Signup and view all the answers

    What aspect of a function does the slope represent in a linear function?

    <p>Rate of change</p> Signup and view all the answers

    In the context of functions, what does 'm' signify in a linear equation?

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

    What is the formula for the inverse of the exponential function $f(x) = 3^x$?

    <p>$f^{-1}(x) = ext{log}_3(x)$</p> Signup and view all the answers

    In a quadratic function, what does the coefficient $a$ determine?

    <p>The concavity of the parabola</p> Signup and view all the answers

    What restriction must be applied to the domain of a quadratic function to ensure its inverse is also a function?

    <p>$x &gt; 0$</p> Signup and view all the answers

    If $f(x) = 4^x$, what would be the inverse of this exponential function?

    <p>$f^{-1}(x) = ext{log}_4(x)$</p> Signup and view all the answers

    What is the key characteristic of a one-to-one function?

    <p>Each element of the domain is associated with exactly one element in the range</p> Signup and view all the answers

    For a linear function, what does the y-intercept represent?

    <p>The value of $y$ when $x = 0$</p> Signup and view all the answers

    What graphical shape represents an exponential function?

    <p>A rapidly increasing or decreasing curve</p> Signup and view all the answers

    What is the significance of the horizontal line test for functions?

    <p>It confirms that a function has an inverse function</p> Signup and view all the answers

    'If no vertical line intersects the graph more than once' describes which test for functions?

    <p>'Vertical Line' Test</p> Signup and view all the answers

    $f(x) = y$ and $f^{-1}(y) = x$ represent which concept in functions?

    <p>'Inverse' Relationship</p> Signup and view all the answers

    What is the inverse of the quadratic function $f(x) = 2x^2$?

    <p>$f^{-1}(x) = \pm \sqrt{2x}$</p> Signup and view all the answers

    For the exponential function $f(x) = 3^x$, what does the horizontal line test specifically help determine about its graph?

    <p>The existence of an inverse function</p> Signup and view all the answers

    In a linear function $f(x) = 2x + 3$, what restriction is placed on the domain for the inverse to exist as a function?

    <p>$x \geq 0$</p> Signup and view all the answers

    For the exponential function $f(x) = a^x$, if $a = 1$, what happens to the graph of the function?

    <p>It becomes a straight line</p> Signup and view all the answers

    Given the linear function $f(x) = 4x - 5$, what would be the result of $f^{-1}(20)$?

    <p>$f^{-1}(20) = 5$</p> Signup and view all the answers

    For a function $f: A \rightarrow B$, what must be true about the function $f$ to ensure that its inverse $f^{-1}$ exists?

    <p>$f$ must be both injective (one-to-one) and surjective (onto)</p> Signup and view all the answers

    Let $f(x) = \log_2(x)$ and $g(x) = 2^x$. Which of the following statements is true?

    <p>$g(x)$ is the inverse function of $f(x)$</p> Signup and view all the answers

    For the function $f(x) = \frac{1}{x^2 - 1}$, what restriction must be placed on the domain to ensure that the inverse function exists?

    <p>$x \neq \pm 1$</p> Signup and view all the answers

    Let $f(x) = \sqrt{x + 5}$. Which of the following statements is true about the inverse function $f^{-1}(x)$?

    <p>$f^{-1}(x) = x^2 - 5$</p> Signup and view all the answers

    Consider the function $f(x) = \tan(x)$. Which of the following statements is true?

    <p>$f(x)$ is injective but not surjective</p> Signup and view all the answers

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