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Section 1.1 : Review : Functions
6. The difference quotient of a 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 \(g\left( x \right) = 6 - {x^2}\).
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Hint : Don’t get excited about the fact that the function is now named \(g\left( x \right)\), the difference quotient still works in the same manner it just has \(g\)’s instead of \(f\)’s now. So, compute \(g\left( {x + h} \right)\), then compute the numerator and finally compute the difference quotient.
\[g\left( {x + h} \right) = 6 - {\left( {x + h} \right)^2} = 6 - {x^2} - 2xh - {h^2}\]
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\[g\left( {x + h} \right) - g\left( x \right) = 6 - {x^2} - 2xh - {h^2} - \left( {6 - {x^2}} \right) = - 2xh - {h^2}\]
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\[\frac{{g\left( {x + h} \right) - g\left( x \right)}}{h} = \frac{{ - 2xh - {h^2}}}{h} = - 2x - h\]