To find the equation of the linear function $f(x)$ passing through the points $(1, 2)$ and $(4, 1)$, we'll use the slope-intercept form of a line:

$f(x) = mx + b$

Step 1: Calculate the Slope $m$

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Plugging in the points $(1, 2)$ and $(4, 1)$: $m = \frac{1 - 2}{4 - 1} = \frac{-1}{3} = -\frac{1}{3}$

Step 2: Use the Slope to Find the y-Intercept $b$

Now that we know $m = -\frac{1}{3}$, we substitute one of the points into the equation $f(x) = mx + b$ to solve for $b$.

Using the point $(1, 2)$: $2 = -\frac{1}{3}(1) + b$ $2 = -\frac{1}{3} + b$ $b = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}$

Step 3: Write the Equation

The equation of the line is: $f(x) = -\frac{1}{3}x + \frac{7}{3}$

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Part (b): Calculate $f(6)$

To find $f(6)$, substitute $x = 6$ into the equation: $f(6) = -\frac{1}{3}(6) + \frac{7}{3}$ $f(6) = -2 + \frac{7}{3}$ Converting $-2$ to a fraction: $f(6) = -\frac{6}{3} + \frac{7}{3} = \frac{1}{3}$

Final Answers

a. The equation for $f(x)$ is: $f(x) = -\frac{1}{3}x + \frac{7}{3}$

b. $f(6) = \frac{1}{3}$

Would you like further details or have questions on any steps?

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Here are 5 related questions for further practice:

1. What is the slope of a line that passes through points $(2, 4)$ and $(5, 7)$?
2. If $f(x) = -\frac{1}{3}x + \frac{7}{3}$, find $f(-3)$.
3. How do you determine if a function is linear by looking at its equation?
4. What is the equation of a line with a slope of $\frac{2}{5}$ that passes through $(0, -3)$?
5. Find the x-intercept and y-intercept of the equation $f(x) = -\frac{1}{3}x + \frac{7}{3}$.

Tip: When finding the slope of a line between two points, remember that the formula is $(y_2 - y_1) / (x_2 - x_1)$ to ensure consistency in direction.