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determine the difference quotient for g(x)=x/(x+3)
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(Level 1) Associate Educator, Expert
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]
The difference quotient is 3/[(x+h+3)(x+3)] or in expanded form, 3/(x^2 + hx + 6x + 3h +9).
Posted by mvcdc on September 4, 2013 at 8:02 PM (Answer #1)
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