To graph the function $g(x) = x^2 - 2x$ for $x \geq 1$ and its inverse, let's proceed step by step.

1. Graphing the Function $g(x) = x^2 - 2x$:

This is a quadratic function. We can rewrite it as: $g(x) = x^2 - 2x = (x - 1)^2 - 1$ This is in vertex form, with the vertex at $(1, -1)$. For $x \geq 1$, this parabola opens upwards and starts from the vertex $(1, -1)$.

Steps to graph:

• Vertex: $(1, -1)$
• Shape: Since the coefficient of $x^2$ is positive, the parabola opens upwards.
• Key Points:
  • At $x = 1$, $g(1) = -1$ (the vertex).
  • At $x = 2$, $g(2) = 0$.
  • At $x = 3$, $g(3) = 3$, and so on.

2. Finding the Inverse:

To find the inverse of $g(x) = x^2 - 2x$ for $x \geq 1$, follow these steps:

1. Start by replacing $g(x)$ with $y$: $y = x^2 - 2x$
2. Solve for $x$ in terms of $y$. First, complete the square on the right-hand side: $y = (x - 1)^2 - 1$ $y + 1 = (x - 1)^2$ $x - 1 = \pm \sqrt{y + 1}$ Since $x \geq 1$, we take the positive square root: $x = 1 + \sqrt{y + 1}$ Therefore, the inverse function is: $g^{-1}(y) = 1 + \sqrt{y + 1}$

3. Graphing the Inverse:

The inverse function $g^{-1}(x) = 1 + \sqrt{x + 1}$ is a square root function that starts at $(x = -1, y = 1)$ and increases as $x$ increases.

Steps to graph:

• Start Point: $(-1, 1)$ (the reflection of the vertex of the original function).
• Shape: The square root function gradually increases as $x$ increases.
• Key Points:
  • At $x = 0$, $g^{-1}(0) = 2$.
  • At $x = 3$, $g^{-1}(3) = 3$, and so on.

Both graphs should be symmetric across the line $y = x$.

Would you like me to graph this for you? Let me know if you need more details.

Here are some related questions to explore:

1. What is the domain and range of $g(x)$ and its inverse?
2. How would the graph of $g(x)$ change if the domain restriction were lifted?
3. Why is it important to consider the domain $x \geq 1$ when finding the inverse of a quadratic function?
4. Can you verify that the inverse function $g^{-1}(x)$ and the original function $g(x)$ are truly inverses algebraically?
5. How does the process of completing the square help in finding the inverse of a quadratic function?

Tip: Always remember that the graph of a function and its inverse are reflections across the line $y = x$.