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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.
Start Solution
$g\left( {x + h} \right) = 6 - {\left( {x + h} \right)^2} = 6 - {x^2} - 2xh - {h^2}$ Show Step 2
$g\left( {x + h} \right) - g\left( x \right) = 6 - {x^2} - 2xh - {h^2} - \left( {6 - {x^2}} \right) = - 2xh - {h^2}$ Show Step 3
$\frac{{g\left( {x + h} \right) - g\left( x \right)}}{h} = \frac{{ - 2xh - {h^2}}}{h} = - 2x - h$