Problem 105 - College Algebra

Question

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

Step-by-Step Solution

Answer

To find the difference quotient of a function \(f\), one needs to substitute \(x+h\) into the function, apply the difference quotient formula and simplify it to its simplest form.

Step 1: Understand the formula and its variables

The formula for the difference quotient is given by \(\frac{f(x+h)-f(x)}{h}\). Here, \(h\) is the difference in \(x\) value, and \(f(x+h)\) and \(f(x)\) are function values at \(x+h\) and \(x\) respectively.

Step 2: Substitute \(x+h\) into the function

Replace every \(x\) in the function \(f\) with \(x+h\). The resulting equation is referred to as \(f(x+h)\), representing the function value at \(x+h\).

Step 3: Apply the difference quotient formula

In the difference quotient formula, replace \(f(x+h)\) and \(f(x)\) with their corresponding function equations, and substitute \(h\) for \(h\) in the denominator.

Step 4: Simplify the difference quotient

Try to simplify the difference quotient to its simplest form. This usually involves expanding brackets in the numerator and simplifying. Cancel out as much as possible. In some cases, one could see terms that would nullify each other, causing the complexity to reduce.