The function $F(x) = 3x + 1$ is a linear function, meaning it represents a straight line when graphed. Let’s analyze and describe it:

Key Points:

1. Slope (m): The coefficient of $x$ is 3, which is the slope of the line. It indicates that for every unit increase in $x$, the value of $F(x)$ increases by 3.
2. Y-intercept (b): The constant term 1 is the y-intercept. It means the line crosses the y-axis at the point $(0, 1)$.
3. General Form: It is already written in the slope-intercept form: $F(x) = mx + b$.
4. Domain: The function is defined for all real numbers, $x \in (-\infty, \infty)$.
5. Range: The output of the function, $F(x)$, is also all real numbers, $F(x) \in (-\infty, \infty)$.

Example Points:

Here are a few points on the line:

• When $x = 0$, $F(0) = 3(0) + 1 = 1$ → Point: $(0, 1)$
• When $x = 1$, $F(1) = 3(1) + 1 = 4$ → Point: $(1, 4)$
• When $x = -1$, $F(-1) = 3(-1) + 1 = -2$ → Point: $(-1, -2)$

Would you like to:

1. Graph this function?
2. Find the inverse of $F(x)$?
3. Solve for $x$ when $F(x) = k$ (specific $k$)?
4. Learn about transformations (shifting/scaling)?
5. Explore real-life applications of this type of function?

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Tip:

Linear functions like $F(x) = 3x + 1$ always graph as straight lines, and their slope determines the steepness of the line!