Problem 73 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=3 x+7$$

Step-by-Step Solution

Answer

The simplified difference quotient for the given function \(f(x)=3x+7\) is \(3\).

Step 1: Find the Function Value at \(x+h\)

The first step is to find the value of the function at \(x+h\), which is denoted \(f(x+h)\). We substitute \(x+h\) into the function \(f(x) = 3x + 7\), which gives: \(f(x+h) = 3(x+h) + 7 = 3x + 3h + 7\).

Step 2: Substitute into the Difference Quotient

The next step is to substitute \(f(x+h)\) and \(f(x)\) into the difference quotient \(\frac{f(x+h)-f(x)}{h}\), which gives: \(\frac{(3x + 3h + 7) - (3x + 7)}{h} = \frac{3h}{h}\). This is obtained by subtracting the terms in the numerator and simplifying.

Step 3: Simplify the Difference Quotient

Now, simplify the difference quotient by cancelling out the \(h\) terms in the numerator and the denominator, which yields: \(\frac{3h}{h} = 3\).