Problem 71 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=4 x$$

Step-by-Step Solution

Answer

The simplified difference quotient for the function \(f(x)=4x\) is \(4\).

Step 1: Substitute the Function into the Formula

Start by substituting \(f(x)=4 x\) into the difference quotient formula, \(\frac{f(x+h)-f(x)}{h}\). So you get \(\frac{4(x+h)-4x}{h}\)

Step 2: Simplify the Equation

Simplify the equation inside the numerator. Distribute \(4\) through the expression \((x+h)\) to get \(4x+4h\). Then you will have \(\frac{4x+4h-4x}{h}\)

Step 3: Cancel out Like Terms

Within the numerator of the difference quotient, \(4x\) and \(-4x\) are like terms and cancel out, leaving us with \(\frac{4h}{h}\)

Step 4: Simplify the Fraction

The fraction \(\frac{4h}{h}\) simplifies to \(4\), because the \(h\) in the numerator and denominator cancel out. Therefore, the difference quotient is \(4\)