Certainly! Let's go through the process of finding and simplifying the difference quotient [tex]\(\frac{F(x + h) - F(x)}{h}\)[/tex] for the function [tex]\(F(x) = \sqrt{x + 9}\)[/tex].
1. Function Evaluation:
We begin by evaluating the function [tex]\(F(x)\)[/tex] and the function at [tex]\(x + h\)[/tex]:
[tex]\[
F(x) = \sqrt{x + 9}
\][/tex]
[tex]\[
F(x + h) = \sqrt{(x + h) + 9} = \sqrt{x + h + 9}
\][/tex]
2. Difference:
Next, we find the difference [tex]\(F(x + h) - F(x)\)[/tex]:
[tex]\[
F(x + h) - F(x) = \sqrt{x + h + 9} - \sqrt{x + 9}
\][/tex]
3. Difference Quotient:
We then form the difference quotient [tex]\(\frac{F(x + h) - F(x)}{h}\)[/tex]:
[tex]\[
\frac{F(x + h) - F(x)}{h} = \frac{\sqrt{x + h + 9} - \sqrt{x + 9}}{h}
\][/tex]
4. Rationalize the Numerator:
To simplify, we rationalize the numerator. We multiply and divide by the conjugate of the numerator:
[tex]\[
\frac{\sqrt{x + h + 9} - \sqrt{x + 9}}{h} \cdot \frac{\sqrt{x + h + 9} + \sqrt{x + 9}}{\sqrt{x + h + 9} + \sqrt{x + 9}}
\][/tex]
This gives us:
[tex]\[
\frac{(\sqrt{x + h + 9} - \sqrt{x + 9})(\sqrt{x + h + 9} + \sqrt{x + 9})}{h (\sqrt{x + h + 9} + \sqrt{x + 9})}
\][/tex]
5. Simplify the Numerator:
Using the difference of squares formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex], the numerator becomes:
[tex]\[
(\sqrt{x + h + 9})^2 - (\sqrt{x + 9})^2 = (x + h + 9) - (x + 9) = x + h + 9 - x - 9 = h
\][/tex]
So the expression simplifies to:
[tex]\[
\frac{h}{h (\sqrt{x + h + 9} + \sqrt{x + 9})}
\][/tex]
6. Cancel [tex]\(h\)[/tex]:
We can now cancel the [tex]\(h\)[/tex] terms in the numerator and denominator:
[tex]\[
\frac{h}{h (\sqrt{x + h + 9} + \sqrt{x + 9})} = \frac{1}{\sqrt{x + h + 9} + \sqrt{x + 9}}
\][/tex]
Thus, the simplified form of the difference quotient is:
[tex]\[
\frac{F(x + h) - F(x)}{h} = \frac{1}{\sqrt{x + h + 9} + \sqrt{x + 9}}
\][/tex]
This completes the process of finding and simplifying the difference quotient for the given function [tex]\(F(x) = \sqrt{x + 9}\)[/tex].
