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In this problem, the task is to determine the equations of three lines: the blue line \( g(x) \), the red line \( h(x) \), and the black line \( f(x) \). The information given is:

1. The red line \( h(x) \) is parallel to the blue line \( g(x) \).
2. The red line \( h(x) \) is perpendicular to the black line \( f(x) \).
3. The points where these lines pass are labeled as follows:
   - \( (0, 4) \) lies on the blue line \( g(x) \).
   - \( (1, 1) \) lies on both the red line \( h(x) \) and the black line \( f(x) \).
   - \( (2, 3) \) lies on the red line \( h(x) \).

### Step 1: Determine the equation of the blue line \( g(x) \)
The blue line passes through the point \( (0, 4) \). Since it is linear, its equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the point \( (0, 4) \), we can directly conclude that the y-intercept is \( 4 \), so \( b = 4 \).

To find the slope \( m \), we need another point on \( g(x) \). Using the fact that the red line is parallel to the blue line, they share the same slope. We can calculate the slope of the red line next and apply it to the blue line.

### Step 2: Determine the slope and equation of the red line \( h(x) \)
The red line passes through the points \( (1, 1) \) and \( (2, 3) \). The slope \( m_h \) is given by:
\[
m_h = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2
\]

So, the equation of the red line \( h(x) \) is in the form:
\[
y = 2x + b
\]
Using the point \( (1, 1) \) to solve for \( b \):
\[
1 = 2(1) + b \implies b = 1 - 2 = -1
\]
Thus, the equation of the red line is:
\[
h(x): y = 2x - 1
\]

### Step 3: Determine the equation of the blue line \( g(x) \)
Since the red line and blue line are parallel, they have the same slope. Therefore, the slope of the blue line \( g(x) \) is also \( m = 2 \). The equation of the blue line is then:
\[
g(x): y = 2x + 4
\]

### Step 4: Determine the equation of the black line \( f(x) \)
The red line is perpendicular to the black line, so the slopes of the two lines must satisfy the relationship:
\[
m_h \cdot m_f = -1
\]
Since \( m_h = 2 \), we have:
\[
2 \cdot m_f = -1 \implies m_f = -\frac{1}{2}
\]
The black line passes through the point \( (1, 1) \), so its equation is:
\[
y = -\frac{1}{2}x + b
\]
Using the point \( (1, 1) \) to solve for \( b \):
\[
1 = -\frac{1}{2}(1) + b \implies 1 = -\frac{1}{2} + b \implies b = 1 + \frac{1}{2} = \frac{3}{2}
\]
Thus, the equation of the black line is:
\[
f(x): y = -\frac{1}{2}x + \frac{3}{2}
\]

### Final Equations:
- Blue line \( g(x) \): \( y = 2x + 4 \)
- Red line \( h(x) \): \( y = 2x - 1 \)
- Black line \( f(x) \): \( y = -\frac{1}{2}x + \frac{3}{2} \)

Would you like further details or explanations?

### Related Questions:
1. How do you find the slope of a line between two points?
2. What is the geometric significance of two lines being perpendicular?
3. Can parallel lines have different y-intercepts, and why?
4. How would you verify if two lines are perpendicular using their slopes?
5. What is the importance of the y-intercept in the equation of a line?

### Tip:
When finding the equation of a line, always start by identifying either two points on the line or a point and the slope.

Determine the equations of three lines: the blue line g(x), the red line h(x), and the black line f(x) given their geometric properties and points they pass through.

Explore how to determine the equations of lines in coordinate geometry. This guide includes the solution for finding the equations of parallel and perpendicular lines through specific points, using slope calculations and linear equations. Ideal for high school students studying geometry.

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