determine the difference quotient for g(x)=x/(x+3)

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mvcdc | Student, Graduate | (Level 2) Associate Educator

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The difference quotient for a function f(x) is the slope of the secant line passing through two points on the curve. We can solve for the difference quotient using the formula:

`[f(x+h) - f(x)]/h`

Our function is `g(x) = x/(x+3)`

The difference quotient is:

`[g(x+h) - g(x)]/h` [Using definition]

`[(x+h)/(x+h+3) - x/(x+3)] /h` [Pluggin in the function]

`[((x+h)(x+3))/((x+h+3)(x+3)) - ((x)(x+h+3))/((x+h+3)(x+3))]/h` [Getting the common denominator]

`[(x+h)(x+3) - x(x+h+3)]/((x+h+3)(x+3)h)` [Simplifying]

`[x^2 + 3x + hx + 3h - x^2 - xh - 3x]/((x+h+3)(x+3)h)` [Expanding]

`(3h)/((x+h+3)(x+3)h)` 

`3/((x+h+3)(x+3))`

The difference quotient is 3/[(x+h+3)(x+3)] or in expanded form, 3/(x^2 + hx + 6x + 3h +9).

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