Problem 74 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=6 x+1$$

Step-by-Step Solution

Answer

The simplified form of the difference quotient for the given function is 6.

Step 1: Substitute the Function Values

Start by substituting the function \(f(x) = 6x + 1\) into the difference quotient formula. So instead of \(f(x+h)\), we plug \(6(x+h)+1\), and instead of \(f(x)\), we use \(6x+1\). The difference quotient becomes \(\frac{6(x+h)+1-(6x+1)}{h}\).

Step 2: Simplify the Expression

Now, simplify the expression in the numerator. The expression becomes \(\frac{6x+6h+1-6x-1}{h}\). Then, \(6x-6x\) and \(1-1\) cancel out, so we are left with \(\frac{6h}{h}\).

Step 3: Simplify Further to Get the Result

Finally, simplify \(\frac{6h}{h}\) which gives a result of 6 - this is as \(h\) cancels out in the numerator and the denominator.