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Questions and Answers
Which of the following represents a closed interval?
Which of the following represents a closed interval?
- [4, 8)
- (−∞, 0)
- [3, 7] (correct)
- (2, 5)
What is the correct notation for an open interval between -2 and 4?
What is the correct notation for an open interval between -2 and 4?
- [−2, 4)
- [-2, 4]
- (−2, 4]
- (-2, 4) (correct)
Which symbol represents that a number can not reach negative infinity?
Which symbol represents that a number can not reach negative infinity?
- −
- ∞
- R
- -∞ (correct)
What does the slope (m) of a line represent?
What does the slope (m) of a line represent?
What is represented by the equation y - y1 = m(x - x1)?
What is represented by the equation y - y1 = m(x - x1)?
What type of interval does [a, ∞) represent?
What type of interval does [a, ∞) represent?
In which type of interval are the endpoints excluded?
In which type of interval are the endpoints excluded?
If the slope of a line is negative, what does that indicate about the line's direction?
If the slope of a line is negative, what does that indicate about the line's direction?
What effect does the transformation y = cf(x) have on the graph of y = f(x)?
What effect does the transformation y = cf(x) have on the graph of y = f(x)?
What type of transformation is applied when using y = -f(x)?
What type of transformation is applied when using y = -f(x)?
How does the transformation y = f(cx) affect the graph of y = f(x)?
How does the transformation y = f(cx) affect the graph of y = f(x)?
What does the transformation y = f(x - 2) accomplish?
What does the transformation y = f(x - 2) accomplish?
Given f(x) = x^2, what is the transformation represented by y = 4f(x)?
Given f(x) = x^2, what is the transformation represented by y = 4f(x)?
What effect does the transformation y = f(-x) have on the graph?
What effect does the transformation y = f(-x) have on the graph?
When applying the transformation y = f(x) + 3, what happens to the graph of y = f(x)?
When applying the transformation y = f(x) + 3, what happens to the graph of y = f(x)?
What outcome results from the transformation y = -2|x| + 3?
What outcome results from the transformation y = -2|x| + 3?
What is the base of the natural logarithm?
What is the base of the natural logarithm?
Which property of logarithms allows you to write $log_a (xy)$ as the sum of $log_a x$ and $log_a y$?
Which property of logarithms allows you to write $log_a (xy)$ as the sum of $log_a x$ and $log_a y$?
Which equation correctly represents the change of base formula?
Which equation correctly represents the change of base formula?
Using the law of logarithms, how can you simplify the expression $ln(x^2)$?
Using the law of logarithms, how can you simplify the expression $ln(x^2)$?
Which of the following represents the logarithmic equation $y = log_a x$ in exponential form?
Which of the following represents the logarithmic equation $y = log_a x$ in exponential form?
If $log_8 (60) - log_8 (3) - log_8 (5)$ is to be simplified using the properties of logarithms, what would be the resultant expression?
If $log_8 (60) - log_8 (3) - log_8 (5)$ is to be simplified using the properties of logarithms, what would be the resultant expression?
If $y = log_a x$, what happens to $x$ as $y$ approaches zero?
If $y = log_a x$, what happens to $x$ as $y$ approaches zero?
Which statement about logarithms is NOT true?
Which statement about logarithms is NOT true?
What is the relationship between the domain of the inverse function and the range of the original function?
What is the relationship between the domain of the inverse function and the range of the original function?
To find an inverse function, which of the following steps is NOT necessary?
To find an inverse function, which of the following steps is NOT necessary?
If f(2) = 5 and f(−3) = 4, what is the value of f −1(5)?
If f(2) = 5 and f(−3) = 4, what is the value of f −1(5)?
Given a function y = f(x), how do you express its inverse?
Given a function y = f(x), how do you express its inverse?
What is the correct expression for the inverse of the function f(x) = 5x^3 + 1?
What is the correct expression for the inverse of the function f(x) = 5x^3 + 1?
Which of the following correctly characterizes the function g(x) = 1 - x^2, x ≤ 1?
Which of the following correctly characterizes the function g(x) = 1 - x^2, x ≤ 1?
What is the expression for f â—¦ g (1) if f(x) = 1, 3, 2, 0 and g(x) = 2, 4, 1, 0?
What is the expression for f â—¦ g (1) if f(x) = 1, 3, 2, 0 and g(x) = 2, 4, 1, 0?
If you want to sketch the graph of the inverse function of f(x), what transformation would you apply?
If you want to sketch the graph of the inverse function of f(x), what transformation would you apply?
What is the value of the number e, approximately?
What is the value of the number e, approximately?
Which of the following describes a one-to-one function?
Which of the following describes a one-to-one function?
What does the horizontal line test determine?
What does the horizontal line test determine?
For the function f(x) = e^x, what is the range?
For the function f(x) = e^x, what is the range?
If f is a one-to-one function, what is the domain of its inverse function f^(-1)?
If f is a one-to-one function, what is the domain of its inverse function f^(-1)?
Which of the following functions is likely to be one-to-one?
Which of the following functions is likely to be one-to-one?
What is the inverse function of f(x) = ax, where a > 0?
What is the inverse function of f(x) = ax, where a > 0?
Which of the following transformations does y = 3 - 2x represent relative to y = 2^x?
Which of the following transformations does y = 3 - 2x represent relative to y = 2^x?
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Study Notes
Set of Numbers
- Natural numbers: N
- Integers: Z
- Rational numbers: Q
- Real numbers: R
Types of Intervals
- Finite intervals:
- Closed interval: [a, b] = {x : a ≤ x ≤ b}
- Open interval: (a, b) = {x : a < x < b}
- Half open/closed interval: (a, b] = {x : a < x ≤ b} or [a, b) = {x : a ≤ x < b}
- Infinite intervals:
- [a, ∞) = {x : x ≥ a}
- (−∞, b) = {x : x < b}
- (−∞, ∞) = R
- Note: ∞ and −∞ are symbols, not real numbers.
Cartesian Plane
- A Cartesian plane is a graph with an x-axis and a y-axis, perpendicular to each other.
- The origin (O) is at the center.
- Positive numbers are to the right (x-axis) and above (y-axis) zero; negative numbers are to the left and below.
Line and Slope
- The equation of a line passing through points (x1, y1) and (x2, y2) is:
- Slope: m = (y2 − y1) / (x2 − x1)
- Equation: y − y1 = m(x − x1)
Transformations of Graphs
- Vertical stretch/shrink:
- y = cf(x): Stretches the graph of y = f(x) vertically by a factor of c (c > 1)
- y = (1/c)f(x): Shrinks the graph of y = f(x) vertically by a factor of c (c > 1)
- Horizontal stretch/shrink:
- y = f(cx): Shrinks the graph of y = f(x) horizontally by a factor of c (c > 1)
- y = f(x/c): Stretches the graph of y = f(x) horizontally by a factor of c (c > 1)
- Reflections:
- y = -f(x): Reflects the graph of y = f(x) about the x-axis
- y = f(-x): Reflects the graph of y = f(x) about the y-axis
The Number e
- e is approximately 2.71828.
The Natural Exponential Function
- f(x) = e^x
One-to-One Functions
- A function f is one-to-one if it never takes on the same value twice: f(x1) ≠f(x2) whenever x1 ≠x2.
- Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.
Inverse Functions
- The inverse function f^-1 of a one-to-one function f has:
- Domain B (range of f)
- Range A (domain of f)
- Definition: f^-1(y) = x ⇔ y = f(x)
- Note: f^-1(x) ≠1/f(x) and [f(x)]^-1 = 1/f(x)
How to Find the Inverse Function
- Write y = f(x).
- Solve for x in terms of y.
- Interchange x and y to get the inverse function: y = f^-1(x).
- The graph of f^-1 is obtained by reflecting the graph of f about the line y = x.
Logarithmic Functions
- General form: y = loga x (a > 0, x > 0)
- Natural logarithmic function: y = ln x = loge x (x > 0)
- Equivalences:
- y = loga x ⇔ x = a^y
- y = ln x ⇔ x = e^y
- Properties:
- loga a^x = x (x ∈ R)
- a^loga x = x (x > 0)
- ln e^x = x (x ∈ R)
- e^ln x = x (x > 0)
Laws of Logarithms
- Product rule: loga (xy) = loga x + loga y and ln(xy) = ln x + ln y
- Quotient rule: loga (x/y) = loga x - loga y and ln(x/y) = ln x - ln y
- Power rule: loga (x^r) = r loga x and ln(x^r) = r ln x
Change of Base Formula
- loga x = ln x / ln a (a > 0, a ≠1)
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