Given the function 
g(x) = 6 − x2,
 evaluate 
g(x + h) − g(x)
h
, h ≠ 0.


To evaluate the expression \(\frac{g(x + h) - g(x)}{h}\) for the function \(g(x) = 6 - x^2\), follow these steps:

### Step 1: Find \(g(x + h)\)
Substitute \(x + h\) into the function \(g(x)\):
\[
g(x + h) = 6 - (x + h)^2
\]

### Step 2: Expand \(g(x + h)\)
Expand the square in the expression:
\[
g(x + h) = 6 - \left(x^2 + 2xh + h^2\right)
\]
Simplify this:
\[
g(x + h) = 6 - x^2 - 2xh - h^2
\]

### Step 3: Find \(g(x + h) - g(x)\)
Subtract \(g(x)\) from \(g(x + h)\):
\[
g(x + h) - g(x) = \left(6 - x^2 - 2xh - h^2\right) - \left(6 - x^2\right)
\]
Simplify the expression by canceling out like terms:
\[
g(x + h) - g(x) = -2xh - h^2
\]

### Step 4: Divide by \(h\)
Now, divide the expression by \(h\):
\[
\frac{g(x + h) - g(x)}{h} = \frac{-2xh - h^2}{h}
\]
Simplify by dividing each term by \(h\):
\[
\frac{g(x + h) - g(x)}{h} = -2x - h
\]

So, the evaluated expression is:
\[
\frac{g(x + h) - g(x)}{h} = -2x - h
\]

Would you like to explore more about this problem or have any questions?

Here are 5 related questions for further exploration:
1. What is the significance of the expression \(\frac{g(x + h) - g(x)}{h}\) in calculus?
2. How would the answer change if \(g(x)\) was a different quadratic function?
3. What happens to the expression as \(h\) approaches 0?
4. How does this result relate to the derivative of \(g(x)\) at \(x\)?
5. What if \(g(x)\) was a cubic function, how would the process differ?

**Tip:** When evaluating expressions like \(\frac{g(x + h) - g(x)}{h}\), you're essentially finding the average rate of change of the function over the interval \(x\) to \(x + h\).

Given the function g(x) = 6 − x^2, evaluate g(x + h) − g(x) / h, h ≠ 0.

This step-by-step guide demonstrates how to evaluate the difference quotient (g(x + h) - g(x)) / h for the quadratic function g(x) = 6 - x^2. Explore key algebraic and calculus concepts, including quadratic expansion and limits, making this suitable for students in Grades 10-12.

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