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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 SolutionWe’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 2Now 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 3We can now finish the problem by computing the full difference quotient.
\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h} = \frac{{ - 9h}}{h} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9}}\]