https://file.aiquickdraw.com/mathbot/file/1727526022945vvs5hdfy.jpg

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 \( y = x \).

Graph g(x) = x^2 - 2x for x ≥ 1, and its inverse function.

Learn how to graph the quadratic function g(x) = x^2 - 2x for x ≥ 1, and its inverse. This problem covers key concepts such as graphing quadratic functions, solving for the inverse, and using the method of completing the square. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation