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Questions and Answers
What is the domain of the function f?
What is the domain of the function f?
- All real numbers less than −1
- All real numbers except 2
- All real numbers
- All real numbers greater than −2/3 (correct)
What is the range of the function f defined in the content?
What is the range of the function f defined in the content?
- y | y ≥ −1 (correct)
- y | y = 0
- y | y ≠0
- y | y < −1
What is the output of the function f(x) = 3 - x when x = 2?
What is the output of the function f(x) = 3 - x when x = 2?
- 3
- 1 (correct)
- 2
- 4
For the function f(x) = 3/(x − 2), which statement about its domain is correct?
For the function f(x) = 3/(x − 2), which statement about its domain is correct?
Which statement correctly describes the domain of the function f(x) = x^2?
Which statement correctly describes the domain of the function f(x) = x^2?
Using set-builder notation, how would you express the domain of the function f(x) = x for nonnegative real numbers?
Using set-builder notation, how would you express the domain of the function f(x) = x for nonnegative real numbers?
What is the resulting equation when solving for x in f(x) = 3?
What is the resulting equation when solving for x in f(x) = 3?
What is the interval notation for the set of real numbers greater than 1 but less than 5?
What is the interval notation for the set of real numbers greater than 1 but less than 5?
Which of the following correctly identifies the range of f(x) = 3/(x − 2)?
Which of the following correctly identifies the range of f(x) = 3/(x − 2)?
When modeling real-world applications, which of the following tools can be used to represent a function?
When modeling real-world applications, which of the following tools can be used to represent a function?
Which function restricts its output to real numbers when the input is nonnegative?
Which function restricts its output to real numbers when the input is nonnegative?
In the temperature function example provided, which variable is the input variable?
In the temperature function example provided, which variable is the input variable?
What is the output set when applying the function f(x) = 3 - x to the domain D = {1, 2, 3}?
What is the output set when applying the function f(x) = 3 - x to the domain D = {1, 2, 3}?
What type of function is associated with the domain of all real numbers?
What type of function is associated with the domain of all real numbers?
Which statement is true regarding the function f(x) = 4 - 2x + 5?
Which statement is true regarding the function f(x) = 4 - 2x + 5?
How can you denote the set of all real numbers between two values a and b, inclusive of a but exclusive of b?
How can you denote the set of all real numbers between two values a and b, inclusive of a but exclusive of b?
What characterizes an even function?
What characterizes an even function?
Which of the following functions is classified as odd?
Which of the following functions is classified as odd?
Which of the following descriptions is true about odd functions?
Which of the following descriptions is true about odd functions?
If f(x) = 2x^5 - 4x + 5, what can be concluded about its symmetry?
If f(x) = 2x^5 - 4x + 5, what can be concluded about its symmetry?
Which of the following statements is NOT true regarding the absolute value function?
Which of the following statements is NOT true regarding the absolute value function?
How is the absolute value function defined for positive inputs?
How is the absolute value function defined for positive inputs?
For a function to be considered even, which of the following must hold true?
For a function to be considered even, which of the following must hold true?
What is the general form of a function that is neither even nor odd?
What is the general form of a function that is neither even nor odd?
What is the value of f(5) for the function defined by f(x) = 3x + 1 for x ≥ 2 and f(x) = x^2 for x < 2?
What is the value of f(5) for the function defined by f(x) = 3x + 1 for x ≥ 2 and f(x) = x^2 for x < 2?
Evaluate f(a + h) for the function f(x) = 3x^2 + 2x - 1.
Evaluate f(a + h) for the function f(x) = 3x^2 + 2x - 1.
What is the domain of the function f(x) = (x - 4)^2 + 5?
What is the domain of the function f(x) = (x - 4)^2 + 5?
What is the minimum value of the function f(x) = (x - 4)^2 + 5?
What is the minimum value of the function f(x) = (x - 4)^2 + 5?
For the function f(x) = 3/(x - 2), what is its domain?
For the function f(x) = 3/(x - 2), what is its domain?
What is the range of the function f(x) = (x - 4)^2 + 5?
What is the range of the function f(x) = (x - 4)^2 + 5?
What is the value of f(-2) for the function f(x) = 3x^2 + 2x - 1?
What is the value of f(-2) for the function f(x) = 3x^2 + 2x - 1?
For the piecewise function f(x) defined as f(x) = 3x + 1 for x ≥ 2, what is f(1)?
For the piecewise function f(x) defined as f(x) = 3x + 1 for x ≥ 2, what is f(1)?
Which of the following represents a function of the form $f(x) = x^2 + 1$?
Which of the following represents a function of the form $f(x) = x^2 + 1$?
What is the y-intercept of the function $f(x) = -x^2$?
What is the y-intercept of the function $f(x) = -x^2$?
Which of the following statements is true regarding the function $f(x) = x^3$?
Which of the following statements is true regarding the function $f(x) = x^3$?
What is the domain of the function $f(x) = x^3$?
What is the domain of the function $f(x) = x^3$?
What can be concluded from the vertical line test for the graph of a function?
What can be concluded from the vertical line test for the graph of a function?
For the function $f(x) = -x^2$, what is the range?
For the function $f(x) = -x^2$, what is the range?
For which intervals is the function $f(x) = x^2 + 1$ increasing?
For which intervals is the function $f(x) = x^2 + 1$ increasing?
Which point is an x-intercept for the function $f(x) = x^3$?
Which point is an x-intercept for the function $f(x) = x^3$?
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Study Notes
Finding Domain & Range
- Domain is the set of all possible input values (x-values) for which a function is defined.
- Range is the set of all possible output values (y-values) that the function can produce.
- To find the domain, consider what values of x would make the function undefined (e.g., division by zero).
- To find the range, consider what values of y can be achieved by the function.
Domain and Range Examples:
- f(x) = (x-4)^2 + 5:
- Domain: All real numbers (represented as (-∞, ∞))
- Range: y ≥ 5 (represented as {y | y ≥ 5})
- f(x) = 3x + 2 - 1:
- Domain: All real numbers
- Range: All real numbers
- f(x) = 3 / (x-2):
- Domain: All real numbers except x = 2 (represented as {x | x ≠2})
- Range: All real numbers except y = 0 (represented as {y | y ≠0})
Representing Functions:
- Functions can be represented using tables, graphs, and formulas.
- Tables: Organize input (x) and output (y) values.
- Graphs: Visualize the function's relationship between input and output, plotting (x, y) points.
- Formulas: Express the function's rule, defining output (y) in terms of input (x).
Natural Domain:
- The domain of a function is considered the natural domain if no specific domain is given.
- This often means all real numbers where the function is defined.
- The square root function f(x) = √x has a natural domain of non-negative real numbers, as the square root of a negative number is not a real number.
Set-builder Notation:
- Defines a set of numbers using a specific property or condition.
- Example: {x | 1 < x < 5} represents the set of all real numbers x where x is greater than 1 and less than 5.
- The notation {x | x has some property} is read as "the set of real numbers x such that x has some property".
Interval Notation:
- Represents a range of numbers.
- Example: (1, 5) represents the interval of all numbers greater than 1 and less than 5 (excluding 1 and 5).
Even and Odd Functions:
- Even Function: f(x) = f(-x) for all x in the domain of f.
- Odd Function: f(-x) = -f(x) for all x in the domain of f.
- Even Functions: Symmetric about the y-axis.
- Odd Functions: Symmetric about the origin (rotational symmetry).
Absolute Value Function:
- Defined as f(x) = -x for x < 0 and f(x) = x for x ≥ 0.
- Represented as |x|.
- Makes all nonzero inputs positive, but outputs 0 if x = 0.
- Range: y ≥ 0 (represented as {y | y ≥ 0}).
- Even function.
Vertical Line Test:
- Determines if a graph represents a function.
- If a vertical line intersects the graph at more than one point, it is not a function.
- If a vertical line intersects the graph at only one point, it is a function.
Function Combinations:
- Sum (f+g): (f+g)(x) = f(x) + g(x)
- Difference (f-g): (f-g)(x) = f(x) - g(x)
- Product (fâ‹…g): (fâ‹…g)(x) = f(x)â‹…g(x)
- Quotient (f/g): (f/g)(x) = f(x) / g(x) (where g(x) ≠0)
- Domain of Combinations: The domain of the new function is the intersection of the domains of f and g, excluding any values that would make the denominator zero in the case of the quotient (f/g).
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