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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<p>$f(f(x)) = \frac{1},{(1/(x+2))^2} = (x+2)^2$ (B)</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>$C(d) = 1000 + 400(d-4)$ (D)</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. (A)</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. (D)</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. (B)</p> Signup and view all the answers

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

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

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

<p>-4 (A)</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 (C)</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 (C)</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) (D)</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) (A)</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 (C)</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 (B)</p> Signup and view all the answers

What is the domain of a rational function?

<p>The set of all real numbers (B)</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 (B)</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 (A)</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 (D)</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. (D)</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>$x = f(y)$ (A)</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>$(f \circ g)(x) = x$ (D)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<p>$f^{-1}(x) = -3x + 4$ (C)</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 (A)</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>$f(x) = \frac{x}{2}$ (C)</p> Signup and view all the answers

What is the domain of a rational function?

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

Flashcards

Rational Function

A function with a polynomial in the numerator and a polynomial in the denominator, where the denominator cannot be zero.

Domain of a Rational Function

The set of all real numbers except for values of 'x' that make the denominator equal to zero.

Asymptote

A line that the graph of a rational function approaches but never intersects.

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

The x-intercept is the point where the graph crosses the x-axis. To find it, set y (or f(x)) equal to zero and solve for x.

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

The y-intercept is the point where the graph crosses the y-axis. To find it, set x equal to zero and solve for y.

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Inverse Function

A function that reverses the effect of another function, meaning if you input a value into a function and then its inverse, you get back the original value.

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Equation for finding the inverse of a function f(x)

To find the inverse function, swap the roles of x and y, then solve for y. This gives you the inverse function, denoted as f⁻¹(x).

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Domain and Range of a Function and its Inverse

The range of a function is the domain of its inverse, and vice versa.

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

To find the inverse, swap x and y and solve for y. In this case, the inverse will be f⁻¹(x) = x + 2.

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Simplify (f + g)(x)

To find (f + g)(x), add the expressions for f(x) and g(x).

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Simplify (f * g)(x)

To find (f * g)(x), multiply the expressions for f(x) and g(x).

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Simplify (f∘g)(x)

To find (f∘g)(x), substitute the expression for g(x) into the function f(x).

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Simplify (f/g)(x)

To find (f/g)(x), divide the expression for f(x) by the expression for g(x).

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Piecewise Function

A function defined by multiple sub-functions, each with its own specific domain.

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Cost of renting a videoke machine

The cost is represented by the piecewise function C(d) = 1000 + 400(d-4), where 'd' is the number of days rented. This function has a base rate of 1000 and an additional charge of 400 for each day after the first 4.

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Cost of hiring guides

The cost of hiring guides is a function of the number of tourists, represented by f(x) = 4/x, where 'x' is the number of tourists. This function shows an inverse relationship - the more tourists, the less the cost per tourist.

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

To find (f + g)(x), add the expressions for f(x) and g(x). In this case, (f + g)(x) = 1/x² + √x.

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

To find (f * g)(x), multiply the expressions for f(x) and g(x). In this case, (f * g)(x) = (x-2)/(x+2).

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

To find (f∘g)(x), substitute the expression for g(x) into the function f(x). In this case, (f∘g)(x) = 1/(x-2).

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(g∘g)(x)

(g∘g)(x) means g(g(x)). This involves plugging the expression for g(x) into the function g(x) again. In this case, it simplifies to: ((√x)-2)+1)/(√x)-2.

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(g∘f)(x)

(g∘f)(x) means g(f(x)). This means plugging the expression for f(x) into the function g(x).

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

(f∘f)(x) means f(f(x)). This involves plugging the expression for f(x) into the function f(x) again. In this case, it simplifies to: (x+2)².

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Simplify (f/g)(x)

To find (f/g)(x), divide the expression for f(x) by the expression for g(x). In this case, (f/g)(x) = x²/(x² + 2x - 2).

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

(f + g)(x) = f(x) + g(x) = 1/(x+2) + (x-2)/x.

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Which of the following best describes a rational function?

A function that has a polynomial in the numerator and a polynomial in the denominator.

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What is the domain of a rational function?

The set of all real numbers except for the values of x that make the denominator zero.

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What is an asymptote in a rational function?

A line that the graph approaches but never intersects.

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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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