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

Solution Steps

To find the difference quotient for the function \( f(x) = x^2 - 9x + 7 \), we need to compute \( \frac{f(x+h) - f(x)}{h} \). First, substitute \( x+h \) into the function to get \( f(x+h) \). Then, subtract \( f(x) \) from \( f(x+h) \) and divide the result by \( h \). Finally, simplify the expression.

Given the function \( f(x) = x^2 - 9x + 7 \), we first calculate \( f(x+h) \) by substituting \( x+h \) into the function:

\[ f(x+h) = (x+h)^2 - 9(x+h) + 7 \]

Expanding this, we get:

\[ f(x+h) = (x^2 + 2xh + h^2) - 9x - 9h + 7 \]

The difference quotient is given by:

\[ \frac{f(x+h) - f(x)}{h} \]

Substituting the expressions for \( f(x+h) \) and \( f(x) \):

\[ \frac{(x^2 + 2xh + h^2 - 9x - 9h + 7) - (x^2 - 9x + 7)}{h} \]

Simplifying the numerator:

\[ = x^2 + 2xh + h^2 - 9x - 9h + 7 - x^2 + 9x - 7 \]

\[ = 2xh + h^2 - 9h \]

Thus, the difference quotient becomes:

\[ \frac{2xh + h^2 - 9h}{h} \]

Factor out \( h \) from the numerator:

\[ = \frac{h(2x + h - 9)}{h} \]

Cancel \( h \) from the numerator and denominator (since \( h \neq 0 \)):

\[ = 2x + h - 9 \]

Final Answer