The domain of a function is the set of all possible x-values (input values). In this case, the x-values are -3, 0, 4, and 11.
Therefore, the correct answer is:
c. {−3, 0, 4, 11}
{(−3, 6), (0, 2), (4, 7), (11, 15)}
a. {(6, −3), (2, 0), (7, 4), (15, 11)}
b. {2, 6, 7, 15}
c. {−3, 0, 4, 11}
d. {−3, 0, 2, 4, 6, 7, 11, 15}
Therefore, the correct answer is:
c. {−3, 0, 4, 11}
a. The graph of y = f(x) will shift left 20 units.
b. The graph of y = f(x) will shift down 20 units.
c. The graph of y = f(x) will shift up 20 units.
d. The graph of y = f(x) will shift right 20 units.
Since the function is being shifted upward, the correct answer is:
c. The graph of y = f(x) will shift up 20 units.
−∞ < y ≤ 3
−3 ≤ y < ∞
0 ≤ y < ∞
3 ≤ y < ∞
Since the absolute value of any number is always non-negative, the range of f(x) will be 0 or greater.
Therefore, the correct answer is:
c. 0 ≤ y < ∞
y = 2x + 16
y = 2x − 20
y equals negative one-half times x
y equals negative one-half times x plus 6
The slope of y = 2x + 3 is 2, so the slope of the perpendicular line will be -1/2.
Using the point-slope form of the equation of a line, we can write the equation of the perpendicular line:
y - y1 = m(x - x1)
where (x1, y1) is the given point (-4, 8) and m is the slope -1/2.
Plugging in the values, we get:
y - 8 = (-1/2)(x - (-4))
y - 8 = (-1/2)(x + 4)
y - 8 = (-1/2)x - 2
y = (-1/2)x - 2 + 8
y = (-1/2)x + 6
Therefore, the correct equation of the line is:
y = (-1/2)x + 6
a. Yes. The input and output are both possible.
b. No. The input is not possible.
c. No. The output is not possible.
d. No. Neither the input nor output is possible.
Typically, the number of friends at the movies is a whole number, as people cannot be in fraction amounts. Therefore, the input of 6.5 does not make sense for this function.
Thus, the correct answer is:
b. No. The input is not possible.
x f(x)
−5 2
−4 1
−3 0
−2 1
−1 2
0 3
1 4
2 5
3 6
What are the vertex and range of the function?
Vertex (−3, 0), Range: {y | 0 ≤ y < ∞}
Vertex (−3, 0), Range: {y | −3 ≤ y < ∞}
Vertex (0, 3), Range: {y | 0 ≤ y < ∞}
Vertex (0, 3), Range: {y | 3 ≤ y < ∞}
In this case, the vertex is (-b/a, c) = (-0/1, 0) = (0, 3).
The range of an absolute value function is the set of all possible y-values (output values). In this case, the range is all values greater than or equal to the y-value at the vertex.
Therefore, the correct answer is:
Vertex (0, 3), Range: {y | 3 ≤ y < ∞}
a line labeled f of x passing through 0, negative 1 and 3, 1
x g(x)
0 2
3 4
6 6
The slope of f(x) is greater than the slope of g(x).
The slope of f(x) is less than the slope of g(x).
The slope of f(x) is equal to the slope of g(x).
The slope of g(x) is undefined.
slope = (change in y)/(change in x)
For f(x), using the points (0, -1) and (3, 1), the slope is:
slope of f(x) = (1 - (-1))/(3 - 0) = 2/3
For g(x), using the points (0, 2) and (3, 4), the slope is:
slope of g(x) = (4 - 2)/(3 - 0) = 2/3
We can see that both linear functions have the same slope, which is 2/3.
Therefore, the correct answer is:
The slope of f(x) is equal to the slope of g(x).
The graph of y = f(x) will shift up 9 units.
The graph of y = f(x) will shift down 9 units.
The graph of y = f(x) will shift left 9 units.
The graph of y = f(x) will shift right 9 units.
When we subtract 9 from x, the effect is a shift of the graph to the right by 9 units.
Therefore, the correct answer is:
The graph of y = f(x) will shift right 9 units.
f(x) = 5x + 100
f(x) = −5x + 100
f(x) = 25x + 75
f(x) = −25x + 75
We are given two points: (5, 75) and (10, 50).
The slope can be calculated using the formula:
slope = (change in y)/(change in x)
slope = (50 - 75)/(10 - 5) = -25/5 = -5
The y-intercept can be found using the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. Plugging in one of the points, (5, 75), we can solve for the y-intercept:
75 = -5(5) + b
75 = -25 + b
75 + 25 = b
b = 100
Therefore, the equation in function form to represent the number of patients seen each week can be written as:
f(x) = -5x + 100
So, the correct answer is:
f(x) = −5x + 100
a. line through the points 0 comma 6 and 12 comma 0
b. line through the points 0 comma negative 6 and 12 comma 0
c. line through the points 0 comma negative 12 and 6 comma 0
d. line through the points 0 comma 12 and 6 comma 0
2x + 4y = 24
4y = -2x + 24
y = (-1/2)x + 6
From the equation, we can see that the slope of the line is -1/2 and the y-intercept is 6.
Now, let's analyze the options:
a. Line through the points (0, 6) and (12, 0)
The points on this line do not match the equation, so this is not the correct graph.
b. Line through the points (0, -6) and (12, 0)
The points on this line do not match the equation, so this is not the correct graph.
c. Line through the points (0, -12) and (6, 0)
The points on this line do not match the equation, so this is not the correct graph.
d. Line through the points (0, 12) and (6, 0)
The points on this line match the equation y = (-1/2)x + 6, so this is the correct graph.
Therefore, the correct answer is:
d. Line through the points (0, 12) and (6, 0)