45 Questions
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
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)
What is the simplified form of (f + g)(x)?
(f + g)(x) = 1/x² + √x
What is the simplified form of (f * g)(x)?
(f * g)(x) = (x-2)/(x+2)
What is the simplified form of (f∘g)(x)?
(f∘g)(x) = 1/(x-2)
Which of the following best represents the function (f - g)(x)?
$\frac{1}{x^2} - \frac{x-2}{x}$
Which of the following best represents the function (f * g)(x)?
$\frac{1}{x+2} \cdot \sqrt{x}$
Which of the following best represents the function (g/f)(x)?
$\frac{\frac{x-2}{x}}{\frac{1}{x+2}}$
Which of the following best represents the function (f∘g)(x)?
$\frac{1}{(x-2)^2}$
Which of the following best represents the function (g∘g)(x)?
$\frac{(\sqrt{x}-2)+1}{\sqrt{x}-2}$
Which of the following best represents the function (f + g)(x)?
$f(x) + g(x) = \frac{1},{x+2} + \frac{x-2},{x}$
What is the simplified form of (f/g)(x)?
$\frac{x^2},{x^2 + 2x - 2}$
Which of the following best represents the function (g∘f)(x)?
$g(f(x)) = \sqrt{x - 2} + 1$
What is the simplified form of (f∘f)(x)?
$f(f(x)) = \frac{1},{(1/(x+2))^2} = (x+2)^2$
Which of the following represents the cost of renting a videoke machine as a piecewise function of the number of days it is rented?
$C(d) = 1000 + 400(d-4)$
Which of the following best describes a rational function?
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.
Which of the following is true about the domain of a rational function?
The domain includes all real numbers except those that make the denominator zero.
Which of the following best describes an asymptote?
A line that approaches the graph of a function but never meets or intersects it.
What is the x-intercept of the rational function f(x) = (x-4)/(x-2)?
4
What is the y-intercept of the rational function f(x) = (x-4)/(x-2)?
-4
Which of the following best describes a rational function?
A function that has a polynomial in the numerator and a polynomial in the denominator
What is the domain of a rational function?
The set of real numbers except for the values of x that make the denominator zero
What is the y-intercept of the rational function $f(x) = \frac{x-4}{x-2}$?
(4, 0)
What is the x-intercept of the rational function $f(x) = \frac{x-4}{x-2}$?
(4, 0)
What is an asymptote in a rational function?
A line that the graph approaches but never intersects
Which of the following best describes a rational function?
A function that has a polynomial in the numerator and a polynomial in the denominator
What is the domain of a rational function?
The set of all real numbers
What is an asymptote in a rational function?
A line that the graph approaches but never intersects
What is the x-intercept of the rational function $f(x) = rac{x-4}{x-2}$?
4
What is the y-intercept of the rational function $f(x) = rac{x-4}{x-2}$?
0
Which of the following best describes an inverse function?
A function that is the mirror image of another function across the line y = x.
Which of the following is the correct equation to find the inverse of a function f(x)?
$x = f(y)$
Which of the following is the correct equation to determine if g is the inverse of f?
$(f \circ g)(x) = x$
What is the inverse of the function f(x) = x - 2?
$f^{-1}(x) = x + 2$
Which of the following is true about the domain and range of a function and its inverse?
The range of a function is the domain of its inverse.
Which of the following functions is the inverse of $f(x) = x^2 - 2x + 3$?
$g(x) = x^2 - 2x + 1$
Which of the following functions is the inverse of $f(x) = \frac{x-2},{x+2}$?
$g(x) = \frac{x+2},{x-2}$
Which of the following functions is the inverse of $f(x) = x^2 + x$?
$g(x) = x^2 - x$
Which of the following functions is the inverse of $f(x) = 1 - 4 - 3x$?
$g(x) = \frac{1},{1 - 4 - 3x}$
Which of the following functions is the inverse of $f(x) = x^2 - 2$?
$g(x) = \sqrt{x - 2}$
Which of the following functions is the inverse of $f(x) = x^2 - 2x + 3$?
$f^{-1}(x) = x^2 - 2x - 3$
Which of the following functions is the inverse of $f(x) = 1 - 4 - 3x$?
$f^{-1}(x) = -3x + 4$
Which of the following best describes a rational function?
A function with a variable in the denominator
Which of the following represents the cost of hiring guides as a function of the number of tourists who wish to explore the cave?
$f(x) = \frac{x}{2}$
What is the domain of a rational function?
All real numbers
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.
Test your understanding of evaluating functions by taking this quiz. Replace the variable in the function with a value from the domain and compute the result.
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