Function: g(x) = 2x2 – 8

For x ≥ 0, the inverse function is f(x)= √
1
2
x + 4

For x ≤ 0, the inverse function is d(x)= – √
1
2
x + 4

Let's analyze the given functions to understand the inverse relationships better.

### Original Function:
The function given is:
\[ g(x) = 2x^2 - 8 \]

### Finding the Inverse:
To find the inverse function, we swap \(x\) and \(y\) in the equation and solve for \(y\).

\[ y = 2x^2 - 8 \]
Swap \(x\) and \(y\):
\[ x = 2y^2 - 8 \]
Now, solve for \(y\):

\[ x + 8 = 2y^2 \]
\[ \frac{x + 8}{2} = y^2 \]
\[ y = \pm \sqrt{\frac{x + 8}{2}} \]

Thus, the inverse function is:
\[ g^{-1}(x) = \pm \sqrt{\frac{x + 8}{2}} \]

### Given Inverse Functions:
The problem states:
- For \( x \geq 0 \), the inverse function is \( f(x) = \sqrt{\frac{1}{2}x + 4} \)
- For \( x \leq 0 \), the inverse function is \( d(x) = -\sqrt{\frac{1}{2}x + 4} \)

### Verification:
Let's simplify and compare the given inverse functions.

1. **For \( x \geq 0 \)**:
   \[ f(x) = \sqrt{\frac{1}{2}x + 4} \]
   This can be rewritten as:
   \[ f(x) = \sqrt{\frac{x}{2} + 4} = \sqrt{\frac{x + 8}{2}} \]

   This matches the positive branch of the inverse we found: 
   \[ g^{-1}(x) = \sqrt{\frac{x + 8}{2}} \]

2. **For \( x \leq 0 \)**:
   \[ d(x) = -\sqrt{\frac{1}{2}x + 4} \]
   This can be rewritten as:
   \[ d(x) = -\sqrt{\frac{x}{2} + 4} = -\sqrt{\frac{x + 8}{2}} \]

   This matches the negative branch of the inverse:
   \[ g^{-1}(x) = -\sqrt{\frac{x + 8}{2}} \]

### Conclusion:
The given inverse functions \( f(x) \) and \( d(x) \) correctly represent the inverse of the original function \( g(x) \) for their respective domains.

### Do you want more details or have any questions?

#### Related Questions:
1. How do you graph the function \( g(x) = 2x^2 - 8 \) and its inverses?
2. What is the domain and range of \( f(x) = \sqrt{\frac{1}{2}x + 4} \)?
3. Can you show the steps for finding the inverse function in more detail?
4. What happens to the inverse function if the domain changes?
5. How can the composition of \( g(x) \) and its inverse be verified?

#### Tip:
When dealing with inverse functions, remember that the domain of the original function becomes the range of the inverse, and vice versa.

Explore the inverse functions of quadratic functions g(x) = 2x^2 - 8 for x ≥ 0 and x ≤ 0. Learn how to find and verify the inverse functions f(x) = √(1/2 x + 4) and d(x) = -√(1/2 x + 4) with detailed steps and examples. Suitable for advanced high school students studying quadratic functions and inverse relationships.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation