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To solve the given problem, we need to analyze the functions \( f(x) \) and \( g(x) \) and answer the questions provided.

### Given Functions:
\[ f(x) = \frac{2}{x} \]
\[ g(x) = x - 1 \]

#### Question 4.1:
Determine the coordinates of the point(s) where the two graphs intersect.

To find the points of intersection, we need to solve the equation:
\[ f(x) = g(x) \]

This translates to:
\[ \frac{2}{x} = x - 1 \]

Multiply both sides by \( x \) to clear the fraction:
\[ 2 = x(x - 1) \]
\[ 2 = x^2 - x \]

Rearrange into a standard quadratic equation form:
\[ x^2 - x - 2 = 0 \]

Factorize the quadratic equation:
\[ (x - 2)(x + 1) = 0 \]

This gives us the solutions:
\[ x = 2 \quad \text{or} \quad x = -1 \]

Now, substitute these values back into either \( f(x) \) or \( g(x) \) to find the corresponding \( y \)-coordinates:

For \( x = 2 \):
\[ y = f(2) = \frac{2}{2} = 1 \]
So, the point is \( (2, 1) \).

For \( x = -1 \):
\[ y = f(-1) = \frac{2}{-1} = -2 \]
So, the point is \( (-1, -2) \).

Therefore, the coordinates of the points of intersection are:
\[ (2, 1) \quad \text{and} \quad (-1, -2) \]

#### Question 4.2:
On the same set of axes, draw the graphs of the two functions. Indicate the coordinates of the point(s) of intersection of the two graphs.

To sketch the graphs:

- The graph of \( f(x) = \frac{2}{x} \) is a hyperbola with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \), respectively.
- The graph of \( g(x) = x - 1 \) is a straight line with a slope of 1 and a y-intercept at \( -1 \).

Indicate the points of intersection \( (2, 1) \) and \( (-1, -2) \) on the graph.

#### Question 4.3:
If \( h(x) = \frac{2}{x + 3} - 4 \), describe the transformation that takes \( f(x) \) to \( g(x) \).

To find the transformation from \( f(x) \) to \( h(x) \):

1. Horizontal shift: \( f(x) \) is horizontally shifted left by 3 units to become \( \frac{2}{x + 3} \).
2. Vertical shift: The resulting function \( \frac{2}{x + 3} \) is then shifted down by 4 units to become \( h(x) = \frac{2}{x + 3} - 4 \).

Therefore, the transformation from \( f(x) \) to \( h(x) \) involves a horizontal shift left by 3 units followed by a vertical shift down by 4 units.

Do you want details or have any questions?

Here are 5 questions you might ask next:
1. How do I find the asymptotes of a rational function?
2. Can you explain the process of factoring quadratic equations?
3. How do I sketch graphs of functions and their transformations?
4. What are the steps to solve a system of equations graphically?
5. Can you help me solve another function-related problem?

**Tip:** When solving for points of intersection between two functions, set the equations equal to each other and solve for the variable, then substitute back to find the corresponding coordinates.

Learn how to solve the intersection of functions f(x) = 2/x and g(x) = x - 1 with a step-by-step solution. This problem covers key concepts in functions, equations, and graphing, including quadratic equation solving. Suitable for Grades 9-12.

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