Problem 80 - 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 \(f(x) = x^{2} - 5x+8\) is \(2x + h - 5\).

Step 1: Substitute the function into the Difference Quotient

Substitute the given function \(f(x) = x^{2}-5 x+8\) into the difference quotient formula. \[ \frac{f(x+h)-f(x)}{h} = \frac{\left[(x + h)^{2}-5 (x + h) +8\right]-\left[(x^{2}-5x+8)\right]}{h} \]

Step 2: Simplify the Numerator

Expand the terms in the numerator and then subtract \((x^{2}-5x+8)\) term. The result is \[ x^{2} + 2hx + h^{2} - 5x - 5h + 8 - x^{2} + 5x - 8 = 2hx + h^{2} - 5h\]

Step 3: Simplify the Difference Quotient by dividing through by h

Divide every term in the numerator by \(h\). This eliminates the \(h\) from the denominator and simplifies the expression: \[ \frac{2hx + h^{2} - 5h}{h} = 2x + h - 5\]