Questions: Find the difference quotient of f; that is, find (f(x+h)-f(x))/h, h ≠ 0, for the following function. Be sure to simplify. f(x)=x^2-4x+9 (f(x+h)-f(x))/h= (Simplify your answer.)

Solution Steps

The difference quotient is given by:

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

First, we need to find \( f(x+h) \). Given \( f(x) = x^2 - 4x + 9 \), substitute \( x+h \) into the function:

\[ f(x+h) = (x+h)^2 - 4(x+h) + 9 \]

Expand the expression for \( f(x+h) \):

\[ f(x+h) = (x+h)^2 - 4(x+h) + 9 = x^2 + 2xh + h^2 - 4x - 4h + 9 \]

Subtract \( f(x) \) from \( f(x+h) \):

\[ f(x+h) - f(x) = (x^2 + 2xh + h^2 - 4x - 4h + 9) - (x^2 - 4x + 9) \]

Simplify the expression:

\[ f(x+h) - f(x) = x^2 + 2xh + h^2 - 4x - 4h + 9 - x^2 + 4x - 9 \]

Cancel out the like terms:

\[ f(x+h) - f(x) = 2xh + h^2 - 4h \]

Now, divide the result by \( h \):

\[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 4h}{h} \]

Factor out \( h \) from the numerator:

\[ \frac{f(x+h) - f(x)}{h} = \frac{h(2x + h - 4)}{h} \]

Cancel \( h \) from the numerator and the denominator:

\[ \frac{f(x+h) - f(x)}{h} = 2x + h - 4 \]

Final Answer