Problem 72 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=7 x$$

Step-by-Step Solution

Answer

The difference quotient simplifies to 7.

Step 1: Substitute for \(f(x+h)\)

Replace every \(x\) in \(f(x)\) with \(x+h\) and calculate the result. So, \(f(x+h) = 7(x+h) = 7x + 7h\).

Step 2: Calculate \(f(x+h)-f(x)\)

Simply subtract the original function \(f(x) = 7x\) from \(f(x+h)\) that we just found. So, \(f(x+h)-f(x) = (7x+7h) - 7x = 7h\).

Step 3: Divide by \(h\)

Take the result from Step 2 and divide by \(h\). So, \(\frac{f(x+h)-f(x)}{h} = \frac{7h}{h}\). Next, perform the division to simplify the expression.

Step 4: Simplify the expression

In terms of \(h\) our difference quotient can be simplified further by cancelling out the \(h\) in the numerator and denominator, which leaves us with \(\frac{7h}{h} = 7\).