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why is (4, 3), (3,5), (3,8), (8,6), (2,14), (4,1) not a function

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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.

what is the y intercept of y=5x-3

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 \((0, -3)\).

what is the slope?

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.

write an equation in point-slope form for the given line. point A is at (1, 5) and point B is at (3, 2)

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.

what is f(3) for the function f(x)=7x+6

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, \(f(3) = 27\).

what are all the steps to solve?

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: \(f(3) = 27\).