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Write a linear function f with the given values

f(0)=2, f(2)=4

3 answers

To write a linear function \( f(x) \) in the form of \( f(x) = mx + b \), we will use the two given points: \( (0, 2) \) and \( (2, 4) \).

  1. Identifying the slope \( m \): The slope \( m \) can be calculated using the formula:

    \[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

    Using the points \( (0, 2) \) and \( (2, 4) \):

    \[ m = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1 \]

  2. Using the slope to find the intercept \( b \): We can substitute one of the points into the linear equation to find \( b \). We'll use the point \( (0, 2) \):

    \[ f(0) = m \cdot 0 + b = 2 \]

    This simplifies to:

    \[ b = 2 \]

  3. Writing the final function: Now we have \( m = 1 \) and \( b = 2 \). Therefore, the linear function is:

    \[ f(x) = 1x + 2 = x + 2 \]

So, the linear function \( f \) that meets the specified conditions is:

\[ f(x) = x + 2 \]

What's the form of y=1x+2 as an answer to that question

The equation \( y = 1x + 2 \) can be simplified as follows:

  1. The coefficient of \( x \) is 1, which can be omitted for simplicity. Thus, we can rewrite it as:

    \[ y = x + 2 \]

In this case, the form is still a linear function, and it adheres to the general linear equation format \( y = mx + b \), where:

  • \( m \) (the slope) is 1,
  • \( b \) (the y-intercept) is 2.

So, in final response to your question, the form of the linear function is:

\[ y = x + 2 \]