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Questions and Answers
What is the range of the function f(x) = cos(x/3)?
What is the range of the function f(x) = cos(x/3)?
- [-1, 1] (correct)
- (-3, 3)
- (-1/3, 1/3)
- (1/3, -1/3)
What is the domain of the function f(x) = sin^{-1}[log2(x/2)]?
What is the domain of the function f(x) = sin^{-1}[log2(x/2)]?
- [-1, 4] (correct)
- None of these
- [1, 4]
- [-4, 1]
Which of these describes the range of the function f(x) = |x + 2|?
Which of these describes the range of the function f(x) = |x + 2|?
- {-1, 1}
- {0, 1}
- R - {-2}
- R (correct)
If f(x) = x + 2, what is f(3)?
If f(x) = x + 2, what is f(3)?
What is the range of the function f(x) = (x^2 + 1)/(x)?
What is the range of the function f(x) = (x^2 + 1)/(x)?
For the function f(x) = x + 2, what will f(0) yield?
For the function f(x) = x + 2, what will f(0) yield?
If f(x) = 3x^2, what is f(1)?
If f(x) = 3x^2, what is f(1)?
What is the property of the graph of an odd function?
What is the property of the graph of an odd function?
What can be concluded if the composition of two functions g and f, written as gof, is one-one?
What can be concluded if the composition of two functions g and f, written as gof, is one-one?
What is a necessary condition for the LCM of a rational number and an irrational number to exist?
What is a necessary condition for the LCM of a rational number and an irrational number to exist?
Which statement about the derivative of an odd function is correct?
Which statement about the derivative of an odd function is correct?
If f(x) is defined as both even and odd, what can be stated about f(x)?
If f(x) is defined as both even and odd, what can be stated about f(x)?
Given two finite sets A and B with m and n elements respectively, how many one-one functions can be formed from A to B if n < m?
Given two finite sets A and B with m and n elements respectively, how many one-one functions can be formed from A to B if n < m?
What does the function f(x) + f(−x) represent if f(x) is any function?
What does the function f(x) + f(−x) represent if f(x) is any function?
If A and B are two different sets with disjoint elements m and n respectively, how many total mappings from A to B can be established?
If A and B are two different sets with disjoint elements m and n respectively, how many total mappings from A to B can be established?
What is the domain of the function f described?
What is the domain of the function f described?
What defines the range of the function f?
What defines the range of the function f?
Which of the following statements is true regarding the outputs of the function f?
Which of the following statements is true regarding the outputs of the function f?
If y = f(x) = 2 - cos(3x), under what condition does y not represent a function of x?
If y = f(x) = 2 - cos(3x), under what condition does y not represent a function of x?
What can be concluded about the relationship between the codomain and the range of function f?
What can be concluded about the relationship between the codomain and the range of function f?
Which of these statements is NOT a requirement for a relation to be considered a function?
Which of these statements is NOT a requirement for a relation to be considered a function?
For the function defined as f(x) = 2 - cos(3x), what is a possible range of f?
For the function defined as f(x) = 2 - cos(3x), what is a possible range of f?
In the function f(x), what does the absence of an image for any input x signify?
In the function f(x), what does the absence of an image for any input x signify?
For the function defined by $f(x) = ext{cos}^2(x) + ext{sin}^4(x)$, what is the range of values for $f(R)$?
For the function defined by $f(x) = ext{cos}^2(x) + ext{sin}^4(x)$, what is the range of values for $f(R)$?
If $f(x) = 2x + 7$ and $g(x) = x^2 + 7$, what values of $x$ satisfy $g(f(x)) = 8$?
If $f(x) = 2x + 7$ and $g(x) = x^2 + 7$, what values of $x$ satisfy $g(f(x)) = 8$?
What transformation does the equation $f(x) = kf(\frac{200x}{100 + x^2})$ imply about $f(x)$?
What transformation does the equation $f(x) = kf(\frac{200x}{100 + x^2})$ imply about $f(x)$?
If $f(x) = x|x|$, which statement about the function $f$ is accurate?
If $f(x) = x|x|$, which statement about the function $f$ is accurate?
Which value of $k$ satisfies the equation $e^{f(x)} = \frac{10 + x}{10 - x}$ for $x \in (-10, 10)$?
Which value of $k$ satisfies the equation $e^{f(x)} = \frac{10 + x}{10 - x}$ for $x \in (-10, 10)$?
When examining the functions $f(x) = 2x + 3$ and $g(x) = x^2 + 7$, which is correct about $g(f(x)) = 8$?
When examining the functions $f(x) = 2x + 3$ and $g(x) = x^2 + 7$, which is correct about $g(f(x)) = 8$?
What type of function is $f(x)$ if $f'(x)$ is even?
What type of function is $f(x)$ if $f'(x)$ is even?
Which range corresponds with the conditions $0 < x < rac{3}{2}$ and $x = y = R$?
Which range corresponds with the conditions $0 < x < rac{3}{2}$ and $x = y = R$?
What kind of function is f defined by the set f = {(1, 1), (2, 1), (3, 0)}?
What kind of function is f defined by the set f = {(1, 1), (2, 1), (3, 0)}?
For which values of x does the equation f(g(x)) = 25 hold, given f(x) = 2x + 3 and g(x) = x^2 + 7?
For which values of x does the equation f(g(x)) = 25 hold, given f(x) = 2x + 3 and g(x) = x^2 + 7?
What is the output of f(2002) if f(x) is defined as f(x) = cos^2(x) + sin^4(x)?
What is the output of f(2002) if f(x) is defined as f(x) = cos^2(x) + sin^4(x)?
What is the domain of the function defined as f(x) = sin^(-1)(log2(x))?
What is the domain of the function defined as f(x) = sin^(-1)(log2(x))?
If f(x) = 1/(x + 2) + 1/(2x - 4) for x > 2, what is f(11)?
If f(x) = 1/(x + 2) + 1/(2x - 4) for x > 2, what is f(11)?
What is the value of f(x) if f(x) is defined as f(x) = cos(log(x))?
What is the value of f(x) if f(x) is defined as f(x) = cos(log(x))?
In the equation f(g(x)) = 25, which mathematical operation does g(x) = x^2 + 7 represent?
In the equation f(g(x)) = 25, which mathematical operation does g(x) = x^2 + 7 represent?
How would you classify f(x) = 2|x| in terms of its behavior?
How would you classify f(x) = 2|x| in terms of its behavior?
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Study Notes
Function Ranges and Definitions
- Range of ( f(x) = \cos(x/3) ) is ([-1, 1]).
- The composition of functions: for ( f: R \to R ) and ( g(x) = x + 3 ), if ( (f \circ g)(x) = (x + 3)^2 ), then ( f(-3) ) can be determined.
Domain of Functions
- Domain of ( f(x) = \sin^{-1}[\log_2(x/2)] ) is defined; among choices: ([1, 4]) is one valid selection.
- Domain of ( f(x) = \sin^{-1}(\log_2 x) ) in real numbers is a set constrained by logarithmic properties.
Composition and Inverses
- If ( f(x) = 2x + 3 ) and ( g(x) = x^2 + 7 ), solving ( g(f(x)) = 8 ) leads to specific values of ( x ).
- Let ( f(x) = x + 3; f(-3) ) seeks output using inverse properties.
Range of Functions
- The range of ( f(x) = \frac{|x + 2|}{x^2 + 1} ) is ( \mathbb{R} ) excluding (-2).
- For ( f(x) = \cos(\log_e x) ), the combined output satisfies specific conditions.
Properties of Functions
- A function is defined as a mapping from set X to Y, connecting each element in X to one element in Y.
- (-1 \leq \cos(3x) \leq 1) ensures the output for ( f(x) ) remains bounded.
Functional Relationships
- If ( f(x) ) is injective (one-to-one), then ( g(f(x)) ) retains that property.
- Functions are defined as even if ( f(x) ) symmetry about the y-axis holds.
Function Transformations
- For transformations, if ( f ) maps real numbers avoiding certain outputs, it affects the domain significantly, constraining areas like ( (−10, 10) ).
Relationships in Sets
- In finite sets ( A ) and ( B ) of different sizes, counting functions can determine how many one-to-one or onto mappings exist, based on element count.
- Special cases exist for functions returning zero, being both odd and even.
Miscellaneous Properties
- Calculations for compositions, such as ( (f \circ f)(x) = x ), require careful definition and function identity.
- Symmetry in graphs aids in determining function properties effectively; odd functions reflect about the origin while even ones reflect about axes.
General Definitions
- ( f(x) + f(-x) ) is even; ( f(x) - f(-x) ) is odd.
- Conditions for ( g \circ f ) to inherit properties depend on the constituent functions being one-to-one or onto.
Practical Application
- Examples illustrate how to derive function outputs through definitions, compositions, consistency in output behavior, and constraints based on real-number properties.
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