General Mathematics - First Semester (1st Periodical)
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General Mathematics - First Semester (1st Periodical)

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Questions and Answers

Which of the following represents the cost of hiring guides as a function of the number of tourists who wish to explore the cave?

  • f(x) = 4/x (correct)
  • f(x) = 4 - x
  • f(x) = 4x
  • f(x) = 4 + x

    • Which of the following represents the cost of renting a videoke machine as a piecewise function of the number of days it is rented?

  • f(x) = 1000 + 400(x-4) (correct)
  • f(x) = 1000 + 400(x-3)
  • f(x) = 1000
  • f(x) = 1000 + 400x

    • What is the simplified form of (f + g)(x)?

  • (f + g)(x) = 1/x² - x + 2
  • (f + g)(x) = 1/x² + √x (correct)
  • (f + g)(x) = 1/x² + x - 2
  • (f + g)(x) = 1/x² - √x

    • What is the simplified form of (f * g)(x)?

    <p>(f * g)(x) = (x-2)/(x+2)</p> Signup and view all the answers

    What is the simplified form of (f∘g)(x)?

    <p>(f∘g)(x) = 1/(x-2)</p> Signup and view all the answers

    Which of the following best represents the function (f - g)(x)?

    <p>$\frac{1}{x^2} - \frac{x-2}{x}$</p> Signup and view all the answers

    Which of the following best represents the function (f * g)(x)?

    <p>$\frac{1}{x+2} \cdot \sqrt{x}$</p> Signup and view all the answers

    Which of the following best represents the function (g/f)(x)?

    <p>$\frac{\frac{x-2}{x}}{\frac{1}{x+2}}$</p> Signup and view all the answers

    Which of the following best represents the function (f∘g)(x)?

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

    Which of the following best represents the function (g∘g)(x)?

    <p>$\frac{(\sqrt{x}-2)+1}{\sqrt{x}-2}$</p> Signup and view all the answers

    Which of the following best represents the function (f + g)(x)?

    <p>$f(x) + g(x) = \frac{1},{x+2} + \frac{x-2},{x}$</p> Signup and view all the answers

    What is the simplified form of (f/g)(x)?

    <p>$\frac{x^2},{x^2 + 2x - 2}$</p> Signup and view all the answers

    Which of the following best represents the function (g∘f)(x)?

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

    What is the simplified form of (f∘f)(x)?

    <p>$f(f(x)) = \frac{1},{(1/(x+2))^2} = (x+2)^2$</p> Signup and view all the answers

    Which of the following represents the cost of renting a videoke machine as a piecewise function of the number of days it is rented?

    <p>$C(d) = 1000 + 400(d-4)$</p> Signup and view all the answers

    Which of the following best describes a rational function?

    <p>A function of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomial functions and q(x) is not equal to zero.</p> Signup and view all the answers

    Which of the following is true about the domain of a rational function?

    <p>The domain includes all real numbers except those that make the denominator zero.</p> Signup and view all the answers

    Which of the following best describes an asymptote?

    <p>A line that approaches the graph of a function but never meets or intersects it.</p> Signup and view all the answers

    What is the x-intercept of the rational function f(x) = (x-4)/(x-2)?

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

    What is the y-intercept of the rational function f(x) = (x-4)/(x-2)?

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

    Which of the following best describes a rational function?

    <p>A function that has a polynomial in the numerator and a polynomial in the denominator</p> Signup and view all the answers

    What is the domain of a rational function?

    <p>The set of real numbers except for the values of x that make the denominator zero</p> Signup and view all the answers

    What is the y-intercept of the rational function $f(x) = \frac{x-4}{x-2}$?

    <p>(4, 0)</p> Signup and view all the answers

    What is the x-intercept of the rational function $f(x) = \frac{x-4}{x-2}$?

    <p>(4, 0)</p> Signup and view all the answers

    What is an asymptote in a rational function?

    <p>A line that the graph approaches but never intersects</p> Signup and view all the answers

    Which of the following best describes a rational function?

    <p>A function that has a polynomial in the numerator and a polynomial in the denominator</p> Signup and view all the answers

    What is the domain of a rational function?

    <p>The set of all real numbers</p> Signup and view all the answers

    What is an asymptote in a rational function?

    <p>A line that the graph approaches but never intersects</p> Signup and view all the answers

    What is the x-intercept of the rational function $f(x) = rac{x-4}{x-2}$?

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

    What is the y-intercept of the rational function $f(x) = rac{x-4}{x-2}$?

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

    Which of the following best describes an inverse function?

    <p>A function that is the mirror image of another function across the line y = x.</p> Signup and view all the answers

    Which of the following is the correct equation to find the inverse of a function f(x)?

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

    Which of the following is the correct equation to determine if g is the inverse of f?

    <p>$(f \circ g)(x) = x$</p> Signup and view all the answers

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

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

    Which of the following is true about the domain and range of a function and its inverse?

    <p>The range of a function is the domain of its inverse.</p> Signup and view all the answers

    Which of the following functions is the inverse of $f(x) = x^2 - 2x + 3$?

    <p>$g(x) = x^2 - 2x + 1$</p> Signup and view all the answers

    Which of the following functions is the inverse of $f(x) = \frac{x-2},{x+2}$?

    <p>$g(x) = \frac{x+2},{x-2}$</p> Signup and view all the answers

    Which of the following functions is the inverse of $f(x) = x^2 + x$?

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

    Which of the following functions is the inverse of $f(x) = 1 - 4 - 3x$?

    <p>$g(x) = \frac{1},{1 - 4 - 3x}$</p> Signup and view all the answers

    Which of the following functions is the inverse of $f(x) = x^2 - 2$?

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

    Which of the following functions is the inverse of $f(x) = x^2 - 2x + 3$?

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

    Which of the following functions is the inverse of $f(x) = 1 - 4 - 3x$?

    <p>$f^{-1}(x) = -3x + 4$</p> Signup and view all the answers

    Which of the following best describes a rational function?

    <p>A function with a variable in the denominator</p> Signup and view all the answers

    Which of the following represents the cost of hiring guides as a function of the number of tourists who wish to explore the cave?

    <p>$f(x) = \frac{x}{2}$</p> Signup and view all the answers

    What is the domain of a rational function?

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

    Study Notes

    Functions and Their Operations

    • The cost of hiring guides can be represented as a function of the number of tourists who wish to explore the cave.
    • The cost of renting a videoke machine can be represented as a piecewise function of the number of days it is rented.

    Composition of Functions

    • The simplified form of (f + g)(x) represents the sum of two functions f and g.
    • The simplified form of (f * g)(x) represents the product of two functions f and g.
    • The simplified form of (f∘g)(x) represents the composition of two functions f and g.

    Inverse Functions

    • The inverse of a function f(x) is denoted as f^(-1)(x) and satisfies the equation f(f^(-1)(x)) = x.
    • The correct equation to find the inverse of a function f is x = f(y).
    • The correct equation to determine if g is the inverse of f is f(g(x)) = g(f(x)) = x.
    • The inverse of a function switches the input and output values.

    Rational Functions

    • A rational function is a function that can be expressed as the ratio of two polynomials.
    • The domain of a rational function is all real numbers except where the denominator is zero.
    • An asymptote in a rational function is a line that the graph approaches as the input values increase or decrease without bound.
    • The x-intercept of a rational function is the point where the graph crosses the x-axis.
    • The y-intercept of a rational function is the point where the graph crosses the y-axis.

    Examples of Functions

    • The inverse of the function f(x) = x - 2 is f^(-1)(x) = x + 2.
    • The inverse of the function f(x) = x^2 - 2x + 3 is a complex function.
    • The inverse of the function f(x) = (x-2)/(x+2) is a complex function.

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