Domain & Range
Zeros
Slope
Intercepts
2024 Pop Culture
100
Determine the domain and range for the below function.
f(x) = 2x + 1
a) First find the domain algebraically: Since this is a linear function with no restrictions, any real number can be input.
b) To find range, notice that for any x value, f(x) will give a real number output.
c) Looking at the graph: The line extends infinitely in both directions without breaks.
Domain: All real numbers (written as (-∞, ∞) or ℝ)
Range: All real numbers ((-∞, ∞) or ℝ)
100
Find the zero of f(x) = 2x + 4
Algebraic Solution:
Set f(x) = 0
2x + 4 = 0
2x = -4
x = -2
Graphical interpretation: The zero occurs where the line intersects the x-axis at (-2, 0).
100
Find the slope of the line passing through points (2,3) and (4,7)
Solution:
* Use slope formula: m = (y₂-y₁)/(x₂-x₁)
* m = (7-3)/(4-2)
* m = 4/2
* m = 2
100
Find the x and y-intercepts of the linear function f(x) = 2x + 4
Algebraic Method:
* For y-intercept: Let x = 0
* f(0) = 2(0) + 4 = 4
* y-intercept is (0, 4)
* For x-intercept: Let f(x) = 0
* 2x + 4 = 0
* 2x = -4
* x = -2
* x-intercept is (-2, 0)
100
What movie was the first to make over $1 billion at the box office in 2024?
"Wonka" starring Timothée Chalamet (which was released in December 2023 but earned much of its box office in early 2024)
200
Determine the domain and range for the below function .
f(x) = -3x + 4 where x ≥ 0
a) Domain is restricted in the question to non-negative numbers
b) For range, substitute the domain restriction:
When x = 0: f(0) = 4
As x increases, f(x) decreases without bound
c) Graph shows a line starting at point (0,4) extending left and down
Domain: [0, ∞)
Range: (-∞, 4]
200
Find the zero of f(x) = -3x + 6
Algebraic Solution:
-3x + 6 = 0
-3x = -6
x = 2
Graphical interpretation: The line crosses the x-axis at (2, 0). Since the coefficient is negative, the line slopes downward from left to right.
200
Find the slope of the line y = -3x + 4
Solution:
* In the slope-intercept form y = mx + b
* -3 is the coefficient of x, which is the slope
* Therefore, m = -3
* The negative slope indicates the line falls 3 units for every 1 unit to the right
200
Find the x and y-intercepts of the linear function f(x) = -3x + 6
Algebraic Method:
* For y-intercept: Let x = 0
* f(0) = -3(0) + 6 = 6
* y-intercept is (0, 6)
* For x-intercept: Let f(x) = 0
* -3x + 6 = 0
* -3x = -6
* x = 2
* x-intercept is (2, 0)
200
Which music artist broke the record for most Grammy wins of all time at the 2024 Grammy Awards?
Taylor Swift, when she won Album of the Year for "Midnights," making it her fourth Album of the Year win
300
Determine the domain and range for the below function f(x) = 2x - 1 where -2 ≤ x ≤ 3
a) Domain is given by the inequality
b) For range, evaluate at endpoints:
At x = -2: f(-2) = -5
At x = 3: f(3) = 5
c) Graph shows a line segment from (-2,-5) to (3,5)
Domain: [-2, 3]
Range: [-5, 5]
300
For the function f(x) = 1/2x - 3, find:
a) The zero of the function
b) The y-intercept
c) Sketch the graph using these points
Algebraic Solution:
Finding zero:
1/2x - 3 = 0
1/2x = 3
x = 6
Y-intercept occurs at x = 0:
f(0) = 1/2(0) - 3 = -3
Graph contains points (6, 0) and (0, -3)
300
In the equation 2x - 3y = 12, find the slope
Solution:
* First, solve for y to get slope-intercept form
* -3y = -2x + 12
* y = (2/3)x - 4
* In y = mx + b form, m = 2/3
* The slope is 2/3, meaning the line rises 2 units for every 3 units to the right
300
Find the x and y-intercepts of the linear function f(x) = (2/3)x - 4
Algebraic Method:
* For y-intercept: Let x = 0
* f(0) = (2/3)(0) - 4 = -4
* y-intercept is (0, -4)
* For x-intercept: Let f(x) = 0
* (2/3)x - 4 = 0
* (2/3)x = 4
* x = 6
* x-intercept is (6, 0)
300
The 2024 Super Bowl LVIII featured which two teams, and who emerged victorious?
The Kansas City Chiefs defeated the San Francisco 49ers, with the Chiefs winning their second consecutive Super Bowl
400
Determine the domain and range for the below function.
f(x) = -x + 2 where -3 < x < 4
a) Domain uses strict inequalities (doesn't include endpoints)
b) For range, evaluate near endpoints:
As x approaches -3: f(x) approaches 5
As x approaches 4: f(x) approaches -2
c) Graph shows a line with open points at (-3,5) and (4,-2)
Domain: (-3, 4)
Range: (-2, 5)
400
Find the zero of f(x) = (4x - 8)/2
Algebraic Solution:
First simplify:
f(x) = 2x - 4
Then solve:
2x - 4 = 0
2x = 4
x = 2
Graphical interpretation: The function is equivalent to 2x - 4, which crosses the x-axis at (2, 0).
400
A line passes through points (-1,-2) and (3,4). Another line is perpendicular to it. What is the slope of the perpendicular line?
Solution:
* First find slope of original line: m₁ = (4-(-2))/(3-(-1)) = 6/4 = 3/2
* For perpendicular lines, m₁ × m₂ = -1
* Therefore, m₂ = -2/3
* The perpendicular line's slope is -2/3
400
Find the x and y-intercepts of the linear function f(x) = (5x - 15)/3
Algebraic Method:
* For y-intercept: Let x = 0
* f(0) = (5(0) - 15)/3 = -5
* y-intercept is (0, -5)
* For x-intercept: Let f(x) = 0
* (5x - 15)/3 = 0
* 5x - 15 = 0
* 5x = 15
* x = 3
* x-intercept is (3, 0)
400
Which highly anticipated video game, a sequel to 2018's "God of War," was released for PC in January 2024?
"God of War Ragnarök"
500
Determine the domain and range for the below piecewise function
f(x) = {2x + 1 for -2 ≤ x < 0
-x + 1 for 0 ≤ x ≤ 3}
a) Domain combines the two intervals
b) For range:
On [-2,0): f(-2) = -3, f(0) = 1
On [0,3]: f(0) = 1, f(3) = -2
c) Graph shows two connected line segments
Domain: [-2, 3]
Range: [-3, 1]
500
If f(x) = mx + b represents a linear function that passes through points (-1, 3) and (2, -3), find:
a) The values of m and b
b) The zero of the function
Algebraic Solution:
a) Using point-slope form:
m = (y₂ - y₁)/(x₂ - x₁)
m = (-3 - 3)/(2 - (-1))
m = -6/3 = -2
Using y = mx + b with (-1, 3):
3 = -2(-1) + b
3 = 2 + b
b = 1
So f(x) = -2x + 1
500
Given the graph below, find the slope of line L:
Solution:
* Identify two points on the line: (0,0) and (3,3)
* Use slope formula: m = (3-0)/(3-0) = 3/3 = 1
* The slope is 1, confirming what we see visually - the line rises at a 45° angle
500
Find the x and y-intercepts of the linear function f(x) = (4x + 8)/-2
Algebraic Method:
* For y-intercept: Let x = 0
* f(0) = (4(0) + 8)/-2 = -4
* y-intercept is (0, -4)
* For x-intercept: Let f(x) = 0
* (4x + 8)/-2 = 0
* 4x + 8 = 0
* 4x = -8
* x = -2
* x-intercept is (-2, 0)
500
Which celebrity couple's relationship made headlines in early 2024, with them being spotted together at multiple NFL games?
Taylor Swift and Travis Kelce of the Kansas City Chiefs