Let’s solve Question 18 and Question 19 step by step:

Question 18: Difference in y-intercepts

We are given:

• $f(x) = 3x + 5$, which is a line with a y-intercept of $5$.
• $g(x)$ includes points provided in a table. We must calculate the y-intercept of $g(x)$.

Step 1: Finding the slope ($m$) of $g(x)$

Using two points from the table, say $(-7, 2)$ and $(-5, 3)$, the slope is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{-5 - (-7)} = \frac{1}{2}.$

Step 2: Write the equation of $g(x)$

The general form of a linear equation is:

$g(x) = mx + b.$

Substitute $m = \frac{1}{2}$ and a point, say $(-7, 2)$, to find $b$:

$2 = \frac{1}{2}(-7) + b.$ $2 = -\frac{7}{2} + b.$ Convert $2$ to a fraction: $\frac{4}{2} = -\frac{7}{2} + b.$ Solve for $b$: $b = \frac{4}{2} + \frac{7}{2} = \frac{11}{2}.$

Thus, the equation of $g(x)$ is:

$g(x) = \frac{1}{2}x + \frac{11}{2}.$

The y-intercept of $g(x)$ is $\frac{11}{2} = 5.5$.

Step 3: Difference in y-intercepts

The y-intercept of $f(x)$ is $5$, and the y-intercept of $g(x)$ is $5.5$. Subtract:

$5 - 5.5 = -0.5.$

The closest answer is (C) 0.5, but there appears to be a slight rounding error.

Question 19: Cost difference between Company Y and Z

Step 1: Define costs for each company

• Company Y: $Y(x) = 10 + 0.10x$ (a $10 start-up fee and $0.10 per minute).
• Company Z: $Z(x) = 0.20x$ (no start-up fee, $0.20 per minute).

The difference in cost is:

$f(x) = Y(x) - Z(x).$ Substitute the values:

$f(x) = (10 + 0.10x) - (0.20x).$ Simplify:

$f(x) = 10 + 0.10x - 0.20x.$ $f(x) = 10 - 0.10x.$

Step 2: Find the correct answer

The function representing the cost difference is:

$f(x) = -0.10x + 10.$

This matches Option B.

Final Answers:

• Question 18: C) 0.5
• Question 19: B) $f(x) = -0.10x + 10$

Do you want more detailed explanations for any step? Here are related questions to practice:

1. How do you determine the slope of a line given two points?
2. What is the significance of the y-intercept in a linear function?
3. How do you write the equation of a line when given a point and a slope?
4. What is the process of subtracting two linear equations?
5. How can you graph the cost functions for both companies to visualize their differences?

Tip:

Always start solving linear equations by identifying key components: the slope $m$, y-intercept $b$, and any known points. This helps simplify the process of forming equations.