Find the difference quotient of $f$ ; that is, find $\frac{f (x + h) - f (x)}{h}, h \neq 0$ , for the following function. Be sure to simplify.
$f (x) = x^{2} - 9 x + 7$
$\frac{f (x + h) - f (x)}{h} = 2 x + h - 9 ($ Simplify your answer.)

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Answer

Final Answer: The difference quotient of the function $f (x) = x^{2} - 9 x + 7$ is $h + 2 x - 9$

Steps

Step 1 ：Given the function $f (x) = x^{2} - 9 x + 7$

Step 2 ：We are asked to find the difference quotient of $f$ , which is $\frac{f (x + h) - f (x)}{h}, h \neq 0$

Step 3 ：Substitute $x + h$ into the function $f (x)$ to get $f (x + h) = (x + h)^{2} - 9 (x + h) + 7$

Step 4 ：Subtract $f (x)$ from $f (x + h)$ to get $f (x + h) - f (x) = (x + h)^{2} - 9 (x + h) + 7 - (x^{2} - 9 x + 7)$

Step 5 ：Simplify the above expression to get $f (x + h) - f (x) = h^{2} + 2 h x - 9 h$

Step 6 ：Divide $f (x + h) - f (x)$ by $h$ to get the difference quotient $\frac{f (x + h) - f (x)}{h} = h + 2 x - 9$

Step 7 ：Final Answer: The difference quotient of the function $f (x) = x^{2} - 9 x + 7$ is $h + 2 x - 9$