Which set represents the domain of the function shown?
{(−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}
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}
Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x) + 20?
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.
The effect of adding 20 to the function will shift the graph vertically.
Since the function is being shifted upward, the correct answer is:
c. The graph of y = f(x) will shift up 20 units.
What is the range of f(x) = |x − 3|?
−∞ < y ≤ 3
−3 ≤ y < ∞
0 ≤ y < ∞
3 ≤ y < ∞
The range of the function f(x) = |x - 3| is the set of all possible y-values (output values).
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 < ∞
What is the equation of the line that is perpendicular to y = 2x + 3 and passes through the point (−4, 8).
y = 2x + 16
y = 2x − 20
y equals negative one-half times x
y equals negative one-half times x plus 6
The line that is perpendicular to y = 2x + 3 will have a slope that is the negative reciprocal of 2.
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 group of friends go to the movies. The function h(x) represents the amount of money spent, where x is the number of friends at the movies. Does a possible solution of (6.5, $95.25) make sense for this function? Explain your answer.
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.
To determine if the solution (6.5, $95.25) makes sense for the function, we need to consider if it is possible for the number of friends to be 6.5.
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.
The table represents an absolute value function f(x).
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 < ∞}
The vertex of an absolute value function in the form f(x) = |ax + b| + c is given by the point (-b/a, c).
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 < ∞}
Compare the slopes of the linear functions f(x) and g(x) and choose the answer that best describes them.
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.
To find the slope of a linear function, we use the formula:
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).
Which of the following statements best describes the effect of replacing the graph of y = f(x) with the graph of y = f(x − 9)?
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.
Replacing the x-values in the function f(x) with x - 9 will result in a horizontal shift of the graph.
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.
A medical clinic is reducing the number of incoming patients by giving vaccines before flu season. During week 5 of flu season, the clinic saw 75 patients. In week 10 of flu season, the clinic saw 50 patients. Assume the reduction in the number of patients each week is linear. Write an equation in function form to show the number of patients seen each week at the clinic.
f(x) = 5x + 100
f(x) = −5x + 100
f(x) = 25x + 75
f(x) = −25x + 75
To find the equation in function form, we need to determine the slope and y-intercept.
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
Choose the graph that correctly represents the equation 2x + 4y = 24.
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
To determine which graph represents the equation 2x + 4y = 24, we can rewrite the equation in slope-intercept form (y = mx + b).
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)