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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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Start Solution

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.

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