The problem asks to find the inverse of the function 
g(x)=x2−12x
 given that 
x
≥
6.


Thank you for the clarification. The problem asks to find the inverse of the function \( g(x) = x^2 - 12x \) given that \( x \geq 6 \).

Let's go through the steps to find the inverse function:

### 1. Set \( g(x) = y \)
\[ y = x^2 - 12x \]

### 2. Swap \( y \) and \( x \) to solve for \( y \)
\[ x = y^2 - 12y \]

### 3. Solve the quadratic equation for \( y \)
We rearrange the equation:
\[ y^2 - 12y - x = 0 \]

This is a quadratic equation in the form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = -12 \), and \( c = -x \).

### 4. Use the quadratic formula to solve for \( y \)
The quadratic formula is:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute \( a = 1 \), \( b = -12 \), and \( c = -x \):
\[ y = \frac{12 \pm \sqrt{(-12)^2 - 4(1)(-x)}}{2(1)} \]
\[ y = \frac{12 \pm \sqrt{144 + 4x}}{2} \]
\[ y = \frac{12 \pm \sqrt{4(36 + x)}}{2} \]
\[ y = \frac{12 \pm 2\sqrt{36 + x}}{2} \]
\[ y = 6 \pm \sqrt{36 + x} \]

### 5. Determine the correct sign
Since \( x \geq 6 \) implies that the function \( g(x) = x^2 - 12x \) is increasing for \( x \geq 6 \), we take the positive square root:
\[ y = 6 + \sqrt{36 + x} \]

### Final Answer
The inverse function is:
\[ g^{-1}(x) = 6 + \sqrt{36 + x} \]

Would you like any further details on this solution? Here are some related questions that might interest you:

1. What is the domain of the inverse function \( g^{-1}(x) \)?
2. How do you graph the inverse function \( g^{-1}(x) \) along with \( g(x) \)?
3. What is the significance of the condition \( x \geq 6 \) in the original function?
4. How would the inverse function change if \( g(x) \) had a different form?
5. Can you verify the inverse function by composing \( g(g^{-1}(x)) \) and \( g^{-1}(g(x)) \)?

**Tip:** When solving for the inverse, ensure that you consider the domain of the original function, as it will influence the correct branch of the square root to use.

Discover how to find the inverse function of g(x) = x^2 - 12x given x ≥ 6. Step-by-step solution with quadratic formula application and domain consideration. Suitable for Grades 10-12 students.

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