To find the value of $b$, we need to determine the equation of the linear function $f(x)$ given the slope and two points on the line.

Step 1: Use the slope-intercept form

The slope-intercept form of a linear equation is: $f(x) = mx + c$ where $m$ is the slope and $c$ is the y-intercept.

Step 2: Substitute the given slope

We are given that the slope $m = 2$, so the equation becomes: $f(x) = 2x + c$

Step 3: Use the point $f(2) = 1$ to find $c$

Substitute $x = 2$ and $f(2) = 1$ into the equation: $1 = 2(2) + c$ $1 = 4 + c$ $c = 1 - 4 = -3$

Step 4: Write the equation of the line

Now that we have $c = -3$, the equation of the line is: $f(x) = 2x - 3$

Step 5: Use $f(b) = 4$ to find $b$

Substitute $f(b) = 4$ into the equation: $4 = 2b - 3$ $2b = 4 + 3$ $2b = 7$ $b = \frac{7}{2} = 3.5$

Conclusion

The value of $b$ is $\boxed{3.5}$.

Would you like further details or have any questions?

Here are some related questions:

1. What is the general form of the equation of a line?
2. How do you find the y-intercept from two points on a line?
3. What is the significance of the slope in the equation of a line?
4. How do you solve for a variable in a linear equation?
5. Can you find $b$ if the slope was different?

Tip: When dealing with linear equations, always remember that the slope remains constant for any two points on the line.