Problem 78 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=x^{2}-5 x+8$$

Step-by-Step Solution

Answer

The simplified difference quotient for the given function is \(2x + h - 5\).

Step 1: Substitute \(f(x+h)\) in the Difference Quotient

First, substitute \(x+h\) into \(f(x) = x^{2} - 5x + 8\). This gives you \(f(x+h) = (x+h)^{2} - 5(x+h) + 8\).

Step 2: Expand and Simplify

Expand \(f(x+h)\) to \(f(x+h) = x^{2} + 2hx + h^{2} - 5x - 5h + 8\). Insert this along with \(f(x)\) into the difference quotient formula, then simplify the numerator by cancelling out the terms which appear on both sides: \(\frac{f(x+h)-f(x)}{h} = \frac{x^{2} + 2hx + h^{2} - 5x - 5h + 8 - (x^{2} - 5x + 8)}{h} = \frac{2hx + h^{2} - 5h}{h}\).

Step 3: Simplify Further

Finally, simplify the difference quotient by eliminating the common factor of \(h\), yielding \(2x + h - 5\).