find difference quotient
f(x) = 1/x+3
recall that the difference quotient is (f(x+h)-f(x))/h
so just plug in your values
(1/(x+h+3) - 1/(x+3))/h
=
((x+3) - (x+h+3))/(x+h+3)(x+3)
---------------------------------------------
h
= (-h)/((x+h+3)(x+3)(h))
= -1/((x+h+3)(x+3)
To find the difference quotient for the function f(x) = 1/(x+3), we will follow these steps:
Step 1: Write down the definition of the difference quotient.
The difference quotient of a function f(x) is defined as:
[f(x+h) - f(x)] / h
Step 2: Substitute the given function f(x) into the formula.
[f(x+h) - f(x)] / h = [1/(x+h+3) - 1/(x+3)] / h
Step 3: Simplify the equation.
To simplify the equation, we need to find a common denominator for the two fractions in the numerator. The common denominator will be (x+h+3)(x+3):
[f(x+h) - f(x)] / h = [(x+3) - (x+h+3)] / h * [(x+h+3)(x+3)] / [(x+h+3)(x+3)]
Simplifying further:
[f(x+h) - f(x)] / h = [x+3 - x - h - 3] / h * [(x+h+3)(x+3)] / [(x+h+3)(x+3)]
[f(x+h) - f(x)] / h = [-h] / h * [(x+h+3)(x+3)] / [(x+h+3)(x+3)]
The h in the numerator and denominator cancels out:
[f(x+h) - f(x)] / h = [-1] * [(x+h+3)(x+3)] / [(x+h+3)(x+3)]
Finally, cancel out the common factors:
[f(x+h) - f(x)] / h = -1
So, the difference quotient for the function f(x) = 1/(x+3) is -1.
To find the difference quotient, you need to compute \(\frac{{f(x + h) - f(x)}}{h}\), where \(f(x)\) is the given function and \(h\) is a small non-zero value. In this case, the given function is \(f(x) = \frac{1}{x+3}\).
Step 1: Substitute \(f(x + h)\) and \(f(x)\) into the formula:
\(\frac{{f(x + h) - f(x)}}{h} = \frac{{\frac{1}{{(x + h) + 3}} - \frac{1}{{x + 3}}}}{h}\).
Step 2: Simplify the expression:
\(\frac{{\frac{1}{{x + h + 3}} - \frac{1}{{x + 3}}}}{h} = \frac{{\frac{{(x + 3) - (x + h + 3)}}{{(x + h + 3)(x + 3)}}}}{h} = \frac{{-h}}{{(x + h + 3)(x + 3)}}\).
So, the difference quotient for the function \(f(x) = \frac{1}{x+3}\) is \(\frac{{-h}}{{(x + h + 3)(x + 3)}}\).