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### Section 3-4 : The Definition of a Function

12. The difference quotient for the function $$f\left( x \right)$$ is defined to be,

$\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$

Compute the difference quotient for the function $$f\left( x \right) = 4 - 9x$$.

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We’ll work this problem in parts. First let’s compute $$f\left( {x + h} \right)$$.

$f\left( {x + h} \right) = 4 - 9\left( {x + h} \right) = 4 - 9x - 9h$ Show Step 2

Now we’ll compute $$f\left( {x + h} \right) - f\left( x \right)$$ and do a little simplification.

$f\left( {x + h} \right) - f\left( x \right) = 4 - 9x - 9h - \left( {4 - 9x} \right) = 4 - 9x - 9h - 4 + 9x = - 9h$

Be careful with the parenthesis when subtracting $$f\left( x \right)$$. We need to subtract the function and so we need parenthesis around the whole thing to make sure we do subtract the function.

Show Step 3

We can now finish the problem by computing the full difference quotient.

$\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h} = \frac{{ - 9h}}{h} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9}}$