Asked by Monkey balls
why is (4, 3), (3,5), (3,8), (8,6), (2,14), (4,1) not a function
Answers
Answered by
Monkey balls
what is the y intercept of y=5x-3
Answered by
Monkey balls
what is the slope?
Answered by
Monkey balls
write an equation in point-slope form for the given line. point A is at (1, 5) and point B is at (3, 2)
Answered by
Monkey balls
what is f(3) for the function f(x)=7x+6
Answered by
Monkey balls
what are all the steps to solve?
Answered by
GPT-4o mini
To determine whether a set of points represents a function, we need to check if each input (in this case, the first coordinate of each ordered pair) is associated with exactly one output (the second coordinate). In other words, a function cannot have two different outputs for the same input.
Let's examine the given points:
1. (4, 3) - input is 4, output is 3
2. (3, 5) - input is 3, output is 5
3. (3, 8) - input is 3, output is 8
4. (8, 6) - input is 8, output is 6
5. (2, 14) - input is 2, output is 14
6. (4, 1) - input is 4, output is 1
Now, let's focus on the inputs:
- The input 4 appears in two pairs: (4, 3) and (4, 1). Since 4 is associated with two different outputs (3 and 1), this violates the definition of a function.
- The input 3 also appears in two pairs: (3, 5) and (3, 8), which means 3 is associated with two outputs (5 and 8).
Because multiple outputs correspond to the same input (4 and 3), the set of points you provided does not define a function.
Let's examine the given points:
1. (4, 3) - input is 4, output is 3
2. (3, 5) - input is 3, output is 5
3. (3, 8) - input is 3, output is 8
4. (8, 6) - input is 8, output is 6
5. (2, 14) - input is 2, output is 14
6. (4, 1) - input is 4, output is 1
Now, let's focus on the inputs:
- The input 4 appears in two pairs: (4, 3) and (4, 1). Since 4 is associated with two different outputs (3 and 1), this violates the definition of a function.
- The input 3 also appears in two pairs: (3, 5) and (3, 8), which means 3 is associated with two outputs (5 and 8).
Because multiple outputs correspond to the same input (4 and 3), the set of points you provided does not define a function.
Answered by
GPT-4o mini
To find the y-intercept of the equation \(y = 5x - 3\), you need to determine the value of \(y\) when \(x = 0\).
Substituting \(x = 0\) into the equation:
\[
y = 5(0) - 3
\]
This simplifies to:
\[
y = 0 - 3 = -3
\]
Therefore, the y-intercept of the equation \(y = 5x - 3\) is \(-3\). In coordinate form, the y-intercept is represented as the point \((0, -3)\).
Substituting \(x = 0\) into the equation:
\[
y = 5(0) - 3
\]
This simplifies to:
\[
y = 0 - 3 = -3
\]
Therefore, the y-intercept of the equation \(y = 5x - 3\) is \(-3\). In coordinate form, the y-intercept is represented as the point \((0, -3)\).
Answered by
GPT-4o mini
The slope of a line in the slope-intercept form of a linear equation, \(y = mx + b\), is represented by \(m\), where:
- \(m\) is the slope,
- \(b\) is the y-intercept.
In the equation \(y = 5x - 3\), the slope \(m\) is \(5\).
Therefore, the slope of the line represented by the equation \(y = 5x - 3\) is \(5\). This means that for every increase of 1 unit in \(x\), \(y\) increases by 5 units.
- \(m\) is the slope,
- \(b\) is the y-intercept.
In the equation \(y = 5x - 3\), the slope \(m\) is \(5\).
Therefore, the slope of the line represented by the equation \(y = 5x - 3\) is \(5\). This means that for every increase of 1 unit in \(x\), \(y\) increases by 5 units.
Answered by
GPT-4o mini
To write the equation of a line in point-slope form, we need the slope of the line and a point on the line. The point-slope form of a line's equation is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
### Step 1: Calculate the slope \(m\)
To find the slope \(m\) between points \(A(1, 5)\) and \(B(3, 2)\), use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
In our case:
- \( (x_1, y_1) = (1, 5) \)
- \( (x_2, y_2) = (3, 2) \)
Substituting in the coordinates:
\[
m = \frac{2 - 5}{3 - 1} = \frac{-3}{2}
\]
### Step 2: Use point-slope form
Now that we have the slope \(m = -\frac{3}{2}\) and we can use point \(A(1, 5)\) for the equation.
Using the point-slope formula:
\[
y - y_1 = m(x - x_1)
\]
Substituting the values:
\[
y - 5 = -\frac{3}{2}(x - 1)
\]
### Final Equation
Thus, the equation in point-slope form is:
\[
y - 5 = -\frac{3}{2}(x - 1)
\]
You can also use point \(B(3, 2)\) to get the same line:
\[
y - 2 = -\frac{3}{2}(x - 3)
\]
Both forms represent the same line.
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
### Step 1: Calculate the slope \(m\)
To find the slope \(m\) between points \(A(1, 5)\) and \(B(3, 2)\), use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
In our case:
- \( (x_1, y_1) = (1, 5) \)
- \( (x_2, y_2) = (3, 2) \)
Substituting in the coordinates:
\[
m = \frac{2 - 5}{3 - 1} = \frac{-3}{2}
\]
### Step 2: Use point-slope form
Now that we have the slope \(m = -\frac{3}{2}\) and we can use point \(A(1, 5)\) for the equation.
Using the point-slope formula:
\[
y - y_1 = m(x - x_1)
\]
Substituting the values:
\[
y - 5 = -\frac{3}{2}(x - 1)
\]
### Final Equation
Thus, the equation in point-slope form is:
\[
y - 5 = -\frac{3}{2}(x - 1)
\]
You can also use point \(B(3, 2)\) to get the same line:
\[
y - 2 = -\frac{3}{2}(x - 3)
\]
Both forms represent the same line.
Answered by
GPT-4o mini
To find \(f(3)\) for the function \(f(x) = 7x + 6\), simply substitute \(3\) for \(x\) in the function:
\[
f(3) = 7(3) + 6
\]
Calculating this:
\[
f(3) = 21 + 6 = 27
\]
Therefore, \(f(3) = 27\).
\[
f(3) = 7(3) + 6
\]
Calculating this:
\[
f(3) = 21 + 6 = 27
\]
Therefore, \(f(3) = 27\).
Answered by
GPT-4o mini
To find \(f(3)\) for the function \(f(x) = 7x + 6\), you can follow these steps in detail:
### Step 1: Understand the Function
The function \(f(x) = 7x + 6\) represents a linear relationship where:
- The coefficient of \(x\) (which is \(7\)) is the slope of the line.
- The constant term (which is \(6\)) is the y-intercept, or the value of \(f(x)\) when \(x = 0\).
### Step 2: Identify the Value to Substitute
You need to find the value of the function when \(x = 3\). This means we will substitute \(3\) for \(x\) in the function.
### Step 3: Substitute the Value into the Function
Replace \(x\) with \(3\) in the equation:
\[
f(3) = 7(3) + 6
\]
### Step 4: Calculate the Multiplication
Perform the multiplication first:
\[
7(3) = 21
\]
### Step 5: Add the Constant
Now, take the result from the multiplication and add \(6\):
\[
21 + 6 = 27
\]
### Step 6: State the Result
Thus, you find that:
\[
f(3) = 27
\]
### Summary of Steps
1. Identify the function: \(f(x) = 7x + 6\).
2. Determine the input value: \(x = 3\).
3. Substitute \(3\) into the function: \(f(3) = 7(3) + 6\).
4. Perform the multiplication: \(7(3) = 21\).
5. Perform the addition: \(21 + 6 = 27\).
6. State the final answer: \(f(3) = 27\).
### Step 1: Understand the Function
The function \(f(x) = 7x + 6\) represents a linear relationship where:
- The coefficient of \(x\) (which is \(7\)) is the slope of the line.
- The constant term (which is \(6\)) is the y-intercept, or the value of \(f(x)\) when \(x = 0\).
### Step 2: Identify the Value to Substitute
You need to find the value of the function when \(x = 3\). This means we will substitute \(3\) for \(x\) in the function.
### Step 3: Substitute the Value into the Function
Replace \(x\) with \(3\) in the equation:
\[
f(3) = 7(3) + 6
\]
### Step 4: Calculate the Multiplication
Perform the multiplication first:
\[
7(3) = 21
\]
### Step 5: Add the Constant
Now, take the result from the multiplication and add \(6\):
\[
21 + 6 = 27
\]
### Step 6: State the Result
Thus, you find that:
\[
f(3) = 27
\]
### Summary of Steps
1. Identify the function: \(f(x) = 7x + 6\).
2. Determine the input value: \(x = 3\).
3. Substitute \(3\) into the function: \(f(3) = 7(3) + 6\).
4. Perform the multiplication: \(7(3) = 21\).
5. Perform the addition: \(21 + 6 = 27\).
6. State the final answer: \(f(3) = 27\).