for the function f(x)=3/(2x-1), determine the difference quotient and simplify
f(x+h)-f(x) = 3/(2(x+h)-1) - 3/(2x-1)
= [3(2x-1) - 3(2x+2h-1)]/[(2x-1)(2(x+h)-1)]
= (6x-3-6x-6h+3)/[(2x-1)(2(x+h)-1)]
= -6h/[(2x-1)(2(x+h)-1)]
now divide that by h to get
-6/[(2x-1)(2(x+h)-1)]
To find the difference quotient for the function f(x) = 3/(2x - 1), we can use the formula:
Difference Quotient = [f(x + h) - f(x)] / h,
where h represents a small change in the x-coordinate.
First, let's find f(x + h). We'll replace x with (x + h) in the given function:
f(x + h) = 3 / (2(x + h) - 1),
Next, we'll subtract f(x) from f(x + h):
[f(x + h) - f(x)] = 3 / (2(x + h) - 1) - 3 / (2x - 1),
To combine these fractions into a single fraction, we'll find a common denominator:
[3(2x - 1) - 3(2(x + h) - 1)] / [(2(x + h) - 1)(2x - 1)],
Simplifying further:
[6x - 3 - 6x - 6h + 3] / [(2(x + h) - 1)(2x - 1)],
Combining like terms:
[-6h] / [(2(x + h) - 1)(2x - 1)],
Finally, dividing both the numerator and denominator by h to obtain the simplified difference quotient:
Difference Quotient = [-6h / h] / [(2(x + h) - 1)(2x - 1) / h],
Difference Quotient = -6 / [(2(x + h) - 1)(2x - 1)].