The image you uploaded contains a mathematical problem involving linear functions. The task is to draw the graph of each function and describe its monotonicity. Let me break down the functions provided and analyze their monotonicity:

a) $f(x) = 1.5x + 0.5$

• Monotonicity: Since the coefficient of $x$ is positive (1.5), this function is increasing. As $x$ increases, $f(x)$ also increases.

b) $f(x) = \frac{3}{4}x - 2$

• Monotonicity: The coefficient of $x$ is positive ($\frac{3}{4}$), so the function is also increasing. As $x$ increases, $f(x)$ increases.

c) $f(x) = 5 - 3x$

• Monotonicity: The coefficient of $x$ is negative (-3), so this function is decreasing. As $x$ increases, $f(x)$ decreases.

d) $f(x) = -1.5x - 0.5$

• Monotonicity: The coefficient of $x$ is negative (-1.5), indicating that this function is decreasing. As $x$ increases, $f(x)$ decreases.

e) $f(x) = -\frac{4}{3}x + 2$

• Monotonicity: The coefficient of $x$ is negative $-\frac{4}{3}$, so this function is decreasing. As $x$ increases, $f(x)$ decreases.

f) $f(x) = -2 + 2x$

• Monotonicity: The coefficient of $x$ is positive (2), so this function is increasing. As $x$ increases, $f(x)$ also increases.

Summary of monotonicity:

• Increasing: a), b), f)
• Decreasing: c), d), e)

Would you like to see a graphical representation of these functions or further explanations on how to plot them? Feel free to ask for any additional details.

5 Related Questions:

1. How do you calculate the slope of a linear function from its equation?
2. What is the significance of the y-intercept in the graph of a linear function?
3. How can you quickly determine if a linear function is increasing or decreasing just by looking at its equation?
4. How do you sketch the graph of a linear function by using two points?
5. What is the general form of a linear function, and how do you recognize it?

Tip: The slope of a linear function indicates its rate of change—if the slope is positive, the function increases, and if it's negative, the function decreases.