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 \( (2, 4) \).
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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 \( (2, 4) \):
\[ m = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1 \]
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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:
\[ b = 2 \]
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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:
\[ f(x) = x + 2 \]