To find the equation of the linear function in the form \( f(x) = mx + b \), we first need to determine the slope \( m \) using the two given points: \( (3, 4) \) and \( (6, 3) \).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in the coordinates of the points:
- \( (x_1, y_1) = (3, 4) \)
- \( (x_2, y_2) = (6, 3) \)
\[ m = \frac{3 - 4}{6 - 3} = \frac{-1}{3} = -\frac{1}{3} \]
Now we have the slope \( m = -\frac{1}{3} \).
Next, we use the point-slope form of a linear equation, which is:
\[ y - y_1 = m(x - x_1) \]
We can use one of the points, for example \( (3, 4) \):
\[ y - 4 = -\frac{1}{3}(x - 3) \]
Now we simplify this equation to the slope-intercept form \( y = mx + b \):
- Distribute \( -\frac{1}{3} \):
\[ y - 4 = -\frac{1}{3}x + 1 \]
- Add 4 to both sides:
\[ y = -\frac{1}{3}x + 1 + 4 \] \[ y = -\frac{1}{3}x + 5 \]
Thus, the equation of the function is:
\[ f(x) = -\frac{1}{3}x + 5 \]
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