Evaluate the function when x=1. f(x)=4x-2. Write the input and output as a set of ordered pairs, and identify the domain and range of the ordered pair. (6 points)
Ordered Pair:
Domain:
Range:
Write an equation of direct variation that passes through the point (9,-1). (4 points)
Write an equation of a line in point-slope form if the slope is -1 and it passes through the point (-3, 5). (4 points)
Write the equation of a line through the point (-2, 1) and perpendicular to 3x+y=1. (4 points)
Write a function rule for the transformation for the graph f(x)=5x.
The graph shifts to the left 4 points, up 2 points, vertically compresses by a factor of 1/2, and reflects in the x-axis. (4 points)
Graph the inequality y<3/2 x+2. (4 points)
So what did you do?
I don't know anything about functions
I need to somehow find the ordered pair, domain, and range
To evaluate the function f(x) = 4x - 2 when x = 1, simply substitute x = 1 into the function:
f(1) = 4(1) - 2
= 4 - 2
= 2
The input-output pair can be written as an ordered pair (1, 2), where the input x is 1 and the output f(x) is 2.
The domain of the ordered pair is the set of possible input values, which in this case can be any real number.
The range of the ordered pair is the set of possible output values, which in this case can also be any real number.
Now, let's move on to the next question.
To write an equation of direct variation that passes through the point (9, -1), we can use the general form of direct variation, which is y = kx. We need to find the value of k, which is the constant of variation.
Substituting the point (9, -1) into the equation, we have:
-1 = 9k
Solving for k, divide both sides by 9:
k = -1/9
Therefore, the equation of direct variation that passes through the point (9, -1) is y = (-1/9)x.
Next question.
To write an equation of a line in point-slope form with a slope of -1 and passing through the point (-3, 5), we can use the point-slope form: y - y1 = m(x - x1).
Using the given slope m = -1 and the point (-3, 5) as (x1, y1), we have:
y - 5 = -1(x - (-3))
y - 5 = -1(x + 3)
y - 5 = -x - 3
y = -x + 2
Therefore, the equation of the line in point-slope form is y = -x + 2.
Moving on.
To find the equation of a line through the point (-2, 1) and perpendicular to 3x + y = 1, we first need to find the slope of the given line by rearranging it into slope-intercept form (y = mx + b).
3x + y = 1
y = -3x + 1
The slope of this line is -3.
Since a line perpendicular to another line has a slope that is the negative reciprocal of the given line's slope, the slope of the perpendicular line is 1/3.
Using the point-slope form, we can write the equation of the perpendicular line:
y - 1 = (1/3)(x - (-2))
y - 1 = (1/3)(x + 2)
y - 1 = (1/3)x + 2/3
y = (1/3)x + 2/3 + 1
y = (1/3)x + 5/3
Therefore, the equation of the line through the point (-2, 1) and perpendicular to 3x + y = 1 is y = (1/3)x + 5/3.
Next question.
To determine the function rule for the transformation of the graph f(x) = 5x, we need to follow the given instructions:
1. Shift to the left 4 points: To shift the function to the left by 4 units, subtract 4 from x inside the function.
f(x) = 5(x - 4)
2. Up 2 points: To shift the function up by 2 units, add 2 to the result of the previous step.
f(x) = 5(x - 4) + 2
3. Vertically compress by a factor of 1/2: To vertically compress the function by a factor of 1/2, multiply the result of the previous step by 1/2.
f(x) = (1/2)(5(x - 4) + 2)
4. Reflect in the x-axis: To reflect the function in the x-axis, negate the entire function.
f(x) = -(1/2)(5(x - 4) + 2)
Simplifying the equation gives:
f(x) = -(5/2)(x - 4) - 1
Therefore, the function rule for the transformation of the graph f(x) = 5x is f(x) = -(5/2)(x - 4) - 1.
Lastly, let's graph the inequality y < (3/2)x + 2.
First, graph the line y = (3/2)x + 2 as a dotted line since it is a strict inequality, meaning the line itself is not included in the solution set.
Choose a test point not on the line, such as (0,0). Substitute these coordinates into the inequality:
0 < (3/2)(0) + 2
0 < 2
Since the inequality is true, shade the region below the line to represent y < (3/2)x + 2.
Mark the line as a dotted line and shade the region below it.
This represents the graph of y < (3/2)x + 2.
At this point, we have answered all the questions. If you have any more questions, feel free to ask!