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Two functions are defined as shown.

f(x) = f(x) equals negative StartFraction 1 Over 2 EndFraction x minus 2. x – 2

g(x) = –1

Which graph shows the input value for which f(x) = g(x)?

1. f(x)
g(x)
x
y
Summary
The image is a coordinate plane with two linear functions plotted. The red line represents the function f(x) and the blue line represents the function g(x). The x-axis and y-axis are labeled, with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The red line (f(x)) has a negative slope, indicating it is a decreasing function. The blue line (g(x)) is horizontal, indicating it is a constant function.
2. f(x)
g(x)
Summary
The image is a graph depicting two functions, ( f(x) ) and ( g(x) ). The function ( f(x) ) is represented by a red line with a negative slope, while ( g(x) ) is represented by a horizontal blue line. The graph is plotted on a Cartesian coordinate system with both the x-axis and y-axis labeled and numbered from -5 to 5. The intersection of the two lines occurs at the point (-2, 1).
3. f(x)
g(x)

Summary
The image is a graph containing two functions plotted on a Cartesian coordinate system. The x-axis and y-axis both range from -5 to 5. The function f(x) is represented by a red line with a negative slope, indicating a linear decreasing function. The function g(x) is represented by a blue horizontal line, indicating a constant function. There is a red dot marking the intersection of the two functions at the point (2, 1).

11 answers

To find where the functions \( f(x) \) and \( g(x) \) intersect, we need to set them equal to each other:

\[ f(x) = g(x) \]

From the problem, we have:

  • \( f(x) = -\frac{1}{2}x - 2 \)
  • \( g(x) = -1 \)

Set the two functions equal:

\[ -\frac{1}{2}x - 2 = -1 \]

Now, solve for \( x \):

  1. Add 2 to both sides: \[ -\frac{1}{2}x = 1 \]

  2. Multiply both sides by -2 to isolate \( x \): \[ x = -2 \]

Now we have found the value of \( x \) at the intersection point \( x = -2 \). To find \( f(-2) \) or \( g(-2) \) to get the corresponding \( y \)-value, we can use either function. Plugging \( x = -2 \) into \( g(x) \):

\[ g(-2) = -1 \]

So, the intersection point is \( (-2, -1) \).

Based on your description of the graphs:

  • The correct graph will show \( f(x) \) and \( g(x) \) intersecting at the point \( (-2, -1) \).

From the summaries you provided, none of them correctly mentioned the intersection point at \( (-2, -1) \). Therefore, none of the provided options correctly depict the input value for which \( f(x) = g(x) \). You should refer back to the graphs you have to look for that specific intersection point, which was likely misdescribed in the summaries.

Two functions are defined as shown.

f(x) = f(x) equals negative StartFraction 1 Over 2 EndFraction x minus 2. x – 2

g(x) = –1

Which graph shows the input value for which f(x) = g(x)?

1. f(x)
g(x)
x
y
Summary
The image is a coordinate plane with two linear functions plotted. The red line represents the function f(x) and the blue line represents the function g(x). The x-axis and y-axis are labeled, with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The red line (f(x)) has a negative slope, indicating it is a decreasing function. The blue line (g(x)) is horizontal, indicating it is a constant function.
2. f(x)
g(x)
Summary
The image is a graph depicting two functions, ( f(x) ) and ( g(x) ). The function ( f(x) ) is represented by a red line with a negative slope, while ( g(x) ) is represented by a horizontal blue line. The graph is plotted on a Cartesian coordinate system with both the x-axis and y-axis labeled and numbered from -5 to 5. The intersection of the two lines occurs at the point (-2, 1).
3. f(x)
g(x)

Summary
The image is a graph containing two functions plotted on a Cartesian coordinate system. The x-axis and y-axis both range from -5 to 5. The function f(x) is represented by a red line with a negative slope, indicating a linear decreasing function. The function g(x) is represented by a blue horizontal line, indicating a constant function. There is a red dot marking the intersection of the two functions at the point (2, 1).

NO there at least needs to one of of the graphs 1 2 and 3 just say which one and ill find out if its right

Based on the descriptions you provided, none of the summaries accurately describe the intersection point where \( f(x) \) equals \( g(x) \). However, if I were to give a suggestion based on proximity to the correct intersection point:

  • Graph 1: This graph states that the red line has a negative slope, indicating it decreases, which is consistent with \( f(x) \).
  • Graph 2: This summary claims the intersection occurs at the point (-2, 1), which is incorrect. The correct intersection point is (-2, -1).
  • Graph 3: This graph suggests the intersection at (2, 1), which is also incorrect.

Since none of the graphs clearly state the intersection as (-2, -1), I would suggest checking Graph 1, as it at least correctly identifies the nature of the functions. However, you might not get a definitive intersection shown there.

Please double-check to see if there is any indication on Graph 1 that suggests it may still illustrate the correct intersection point on the coordinate plane.

OHHH WAIT THERES ONE MORE f(x)
g(x)

Summary
The image is a Cartesian coordinate graph with two functions plotted on it. The red line represents the function f(x), which is a decreasing linear function. The blue line represents the function g(x), which is a constant function. The graph shows the intersection of these two lines at a point where f(x) equals g(x). The x-axis and y-axis are labeled, and the grid is visible to show the values of x and y coordinates.

is this the one? or still 1

Based on the new summary you provided, which states that the graph shows the intersection of the two lines at a point where \( f(x) \) equals \( g(x) \), it suggests that this graph accurately depicts the intersection point.

If this new graph clearly identifies the intersection point between the red line (representing \( f(x) \)) and the blue line (representing \( g(x) \)), then this is likely the correct graph showing where \( f(x) = g(x) \).

If the intersection point is indeed correctly represented in the new graph, then I would recommend choosing this one as the correct answer. Please verify that it accurately depicts the intersection at \( (-2, -1) \) to confirm it's the correct option.

Mustafa’s soccer team is planning a school dance as a fundraiser. The DJ charges $200 and decorations cost $100. The team decides to charge each student $5.00 to attend the dance. If n represents the number of students attending the dance, which equation can be used to find the number of students needed to make $1,500 in profit?

5n – 300 = 1,500
5n + 300 = 1,500
5n – 200 + 100n = 1,500
5n – 100 – 200n = 1,500

To determine the equation that represents the situation described, we first need to calculate the total costs and the revenue generated from the students attending the dance.

Costs:

  • DJ charges: $200
  • Decorations cost: $100
  • Total costs = $200 + $100 = $300

Revenue:

  • Each student pays $5 to attend, so for \( n \) students attending, the revenue is \( 5n \).

Profit:

  • Profit is defined as total revenue minus total costs. We want to find the number of students needed for the profit to be $1,500.

The equation for profit can be set up as follows:

\[ \text{Profit} = \text{Revenue} - \text{Costs} \]

Substituting in the values we have:

\[ 1,500 = 5n - 300 \]

This can be rearranged to form the equation:

\[ 5n - 300 = 1,500 \]

Thus, the correct equation that can be used to find the number of students needed to make $1,500 in profit is:

5n – 300 = 1,500

The equation 9(u – 2) + 1.5u = 8.25 models the total miles Michael traveled one afternoon while sledding, where u equals the number of hours walking up a hill and (u – 2) equals the number of hours sledding down the hill. Which is the value of u?

u = 0.25
u = 0.75
u = 1.1
u = 2.5

To find the value of \( u \) in the equation \( 9(u - 2) + 1.5u = 8.25 \), we will solve it step by step.

  1. Distribute the 9: \[ 9(u - 2) = 9u - 18 \] So the equation becomes: \[ 9u - 18 + 1.5u = 8.25 \]

  2. Combine like terms: \[ (9u + 1.5u) - 18 = 8.25 \] This simplifies to: \[ 10.5u - 18 = 8.25 \]

  3. Add 18 to both sides: \[ 10.5u = 8.25 + 18 \] \[ 10.5u = 26.25 \]

  4. Divide both sides by 10.5: \[ u = \frac{26.25}{10.5} \]

  5. Calculate \( u \): \[ u = 2.5 \]

Thus, the value of \( u \) is \( \boxed{2.5} \).

Rahul solved the equation 2(x – ) – 2 left-parenthesis x minus StartFraction 1 Over 8 EndFraction right-parenthesis minus StartFraction 3 Over 5 EndFraction x equals StartFraction 55 Over 4 EndFraction x = 2 left-parenthesis x minus StartFraction 1 Over 8 EndFraction right-parenthesis minus StartFraction 3 Over 5 EndFraction x equals StartFraction 55 Over 4 EndFraction . In which step did he use the addition property of equality?

step 1 2 3 or 4?

Rahul’s Solution

Steps Resulting equations
1 2x - 1/4 - 3/5x = 55/4
2 7/5x - 1/4 = 55/4
3 7/5x = 56/4
4 x = 10
Summary
The image shows a step-by-step solution to an equation. Rahul starts with the equation (2x - \frac{1}{4} - \frac{3}{5}x = \frac{55}{4}), and through a series of steps, simplifies it to find that (x = 10).

To determine in which step Rahul used the addition property of equality, we need to understand what that property states. The addition property of equality states that if two expressions are equal, adding the same value to both sides of the equation maintains the equality.

Let's examine each step in Rahul's solution:

  1. Step 1: \( 2x - \frac{1}{4} - \frac{3}{5}x = \frac{55}{4} \)
    This is the original equation and no addition has been done yet.

  2. Step 2: \( \frac{7}{5}x - \frac{1}{4} = \frac{55}{4} \)
    This step combines \( 2x \) and \(-\frac{3}{5}x\). It does not involve adding a term to both sides.

  3. Step 3: \( \frac{7}{5}x = \frac{56}{4} \)
    In this step, Rahul added \(\frac{1}{4}\) to both sides. This uses the addition property of equality.

  4. Step 4: \( x = 10 \)
    This step involves dividing both sides by \(\frac{7}{5}\) and does not involve addition.

So, the addition property of equality was used in Step 3.