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Questions and Answers
Solve −4(2x − 1) = −6(x + 2) − 2.
Solve −4(2x − 1) = −6(x + 2) − 2.
9
Rearrange the formula C = 2Ï€r to solve for r.
Rearrange the formula C = 2Ï€r to solve for r.
r = C/2Ï€
Solve −2x − 13 = 8x + 7.
Solve −2x − 13 = 8x + 7.
-2
Solve 4x − 9 = 15.
Solve 4x − 9 = 15.
Given the equation 3x − 3 = 6x + 1, select the reasoning that correctly solves for x.
Given the equation 3x − 3 = 6x + 1, select the reasoning that correctly solves for x.
Solve x/4 = 8.
Solve x/4 = 8.
Solve −5x = 30.
Solve −5x = 30.
In order to solve the equation −3x + 6 = 5, what is the first step?
In order to solve the equation −3x + 6 = 5, what is the first step?
Let f(x) = x^2 − 3x − 7. Find f(−3).
Let f(x) = x^2 − 3x − 7. Find f(−3).
Let f(x) = 5x + 12. Find f^−1(x).
Let f(x) = 5x + 12. Find f^−1(x).
Let f(x) = 12/4 x + 2. Find f(−1).
Let f(x) = 12/4 x + 2. Find f(−1).
Let f(x) = 9x − 2 and g(x) = −x + 3. Find f(g(x)).
Let f(x) = 9x − 2 and g(x) = −x + 3. Find f(g(x)).
Let f(x) = x − 2 and g(x) = x^2 − 7x − 9. Find f(g(−1)).
Let f(x) = x − 2 and g(x) = x^2 − 7x − 9. Find f(g(−1)).
Let f(x) = 2x − 6. Solve f^−1(x) when x = 2.
Let f(x) = 2x − 6. Solve f^−1(x) when x = 2.
On the graph of the equation x - 4y = -16, what is the value of the y-intercept?
On the graph of the equation x - 4y = -16, what is the value of the y-intercept?
For the functions f(x) = 2x + 2 and g(x) = 7x + 1, which composition produces the greatest output?
For the functions f(x) = 2x + 2 and g(x) = 7x + 1, which composition produces the greatest output?
Which of the following expressions represents a function: {(1, 2), (4, -2), (8, 3), (9, -3)}?
Which of the following expressions represents a function: {(1, 2), (4, -2), (8, 3), (9, -3)}?
Consider the function f(x) = 8 − 3x^2. Find the value of f(x) when x = 3.
Consider the function f(x) = 8 − 3x^2. Find the value of f(x) when x = 3.
Given the function f(x) = 3x + 4, which of the following expressions is the inverse?
Given the function f(x) = 3x + 4, which of the following expressions is the inverse?
A.Given the equation 2(3x − 4) = 5x + 6, solve for the variable. What is Charlie's solution?
A.Given the equation 2(3x − 4) = 5x + 6, solve for the variable. What is Charlie's solution?
If the linear function representing the amount of rainfall each week is f(w) = 2.5w + 3, what does the 3 represent in this function?
If the linear function representing the amount of rainfall each week is f(w) = 2.5w + 3, what does the 3 represent in this function?
What type of line is the graph of 4x + y = −3?
What type of line is the graph of 4x + y = −3?
What is the slope of the line between (3, −4) and (−2, 1)?
What is the slope of the line between (3, −4) and (−2, 1)?
What is the slope of the line y = 3?
What is the slope of the line y = 3?
What is the slope of the line between (-4, 4) and (-1, -2)?
What is the slope of the line between (-4, 4) and (-1, -2)?
What is the slope of the line that passes through (2, 12) and (4, 20)?
What is the slope of the line that passes through (2, 12) and (4, 20)?
What is the equation of a horizontal line passing through the point (−7, 5)?
What is the equation of a horizontal line passing through the point (−7, 5)?
Given the equation 4x − 26 = 2x − 4, what order of operations completely solves for x?
Given the equation 4x − 26 = 2x − 4, what order of operations completely solves for x?
Which function has the largest slope?
Which function has the largest slope?
Given that f(x) = 2x + 1 and g(x) = −5x + 2, solve for f(g(x)) when x = 3.
Given that f(x) = 2x + 1 and g(x) = −5x + 2, solve for f(g(x)) when x = 3.
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Study Notes
Key Equations and Concepts
- Circumference of a Circle: C = 2Ï€r; rearranged to solve for radius: r = C/2Ï€.
- Common Linear Equation Forms:
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
- Point-Slope Form: y - y1 = m(x - x1), used when a point and slope are known.
Solving Linear Equations
- Typical solving strategy: isolate the variable through addition, subtraction, multiplication, or division.
- Example: -4(2x − 1) = −6(x + 2) − 2 results in x = 9.
- Example: From -2x − 13 = 8x + 7, deduced x = -2.
Functions and Their Inverses
- For a function f(x) = ax + b, the inverse process involves swapping x and y, leading to f^−1(x).
- Example: f(x) = 2x + 2; solving f^−1(x) gives x = 4.
- Composition of functions: f(g(x)) requires evaluating the outer function at the result of the inner function.
Graphing and Slopes
- Slope Calculation: The slope between two points (x1, y1) and (x2, y2) is m = (y2 - y1) / (x2 - x1).
- Example: Slope between (3, -4) and (-2, 1) is -1.
- Horizontal line has a slope of 0; vertical line is undefined.
Linear Models in Context
- Example: A function representing rainfall f(w) = 2.5w + 3 indicates that 3 is the baseline rainfall before week 1.
- Example: Alex saves money modeled by M = 6h + 2, denoting initial savings and rate of saving.
Errors and Reasoning in Problem Solving
- Identifying mistakes in calculations is crucial. Example: Aaron erred by subtracting incorrectly in solving the equation.
- Understanding the order of operations and justifying each step is essential for accurate results.
Special Function Characteristics
- Functions like f(x) = x^2 + 3x + 6 can yield specific outputs for given x-values through substitution.
- Ensure that the range for practical functions, such as the number of text messages sent, only includes integers.
Application Problems
- Real-world problems, like Joe's earnings model, combine fixed rates and variable conditions to derive total outcomes.
- A fish tank model (F = 0.4P) helps quantify relationships between variables such as fish count and plant presence.
Visual Characteristics of Graphs
- A falling line indicates a negative slope, such as x = -5, which is a vertical line.
- The y-intercept is often determined by substituting x = 0 in the equation.
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