Problem 124 - Precalculus Essentials

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 \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\), substitute \(f(x+h)\) and \(f(x)\) with the provided function, simplify the numerator, and then cancel out any common factors in the numerator and denominator.

Step 1: Identify the Function \(f\)

Before you can calculate the difference quotient, you need to know the function \(f\). Make sure this has been given or that you can find it from the information available.

Step 2: Substitute Function into Difference Quotient

Replace \(f(x+h)\) and \(f(x)\) in the difference quotient formula \(\frac{f(x+h)-f(x)}{h}\) with the actual function \(f\). This means that wherever in the function there’s an \(x\), swap it out for \((x+h)\) to find \(f(x+h)\), and keep \(f(x)\) as it is.

Step 3: Simplify

Simplify the expression in the numerator, \(f(x+h)-f(x)\). This will involve expanding any brackets and combining any like terms. If the function \(f\) is a polynomial, the \(h\) in some terms should cancel out during this step.

Step 4: Cancel out Common Factors

Cancel out any common factors in the numerator and the denominator (which in this case is \(h\)). The result will be your difference quotient of the function \(f\).