Question

Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{1-2 x}$$

Short answer

Question: Simplify the following difference quotients for the function \(f(x) = \sqrt{1-2x}\): 1. \(\frac{f(x+h)-f(x)}{h}\) 2. \(\frac{f(x)-f(a)}{x-a}\) Answer: 1. \(\frac{f(x+h)-f(x)}{h} = \frac{-2}{\sqrt{1-2x-2h} + \sqrt{1-2x}}\) 2. \(\frac{f(x)-f(a)}{x-a} = \frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}\)

Step-by-step solution

Find \(f(x+h)\) and \(f(a)\)

Calculate the value of the function at \(x+h\) and \(a\): $$f(x+h) = \sqrt{1-2(x+h)} = \sqrt{1-2x-2h}$$ $$f(a) = \sqrt{1-2a}$$

Form the first difference quotient

Form the difference quotient \(\frac{f(x+h)-f(x)}{h}\): $$\frac{f(x+h)-f(x)}{h} = \frac{\sqrt{1-2x-2h} - \sqrt{1-2x}}{h}$$

Rationalize the first difference quotient

Rationalize the first difference quotient by multiplying both numerator and denominator by the conjugate of the numerator: $$\frac{\sqrt{1-2x-2h} - \sqrt{1-2x}}{h} \cdot \frac{\sqrt{1-2x-2h} + \sqrt{1-2x}}{\sqrt{1-2x-2h} + \sqrt{1-2x}} = \frac{ (1-2x-2h)-(1-2x)}{(h)(\sqrt{1-2x-2h} + \sqrt{1-2x})}$$

Simplify the first difference quotient

Simplify the first difference quotient: $$\frac{-2h}{h(\sqrt{1-2x-2h} + \sqrt{1-2x})} = \frac{-2}{\sqrt{1-2x-2h} + \sqrt{1-2x}}$$ Now let's simplify the second difference quotient:

Form the second difference quotient

Form the difference quotient \(\frac{f(x)-f(a)}{x-a}\): $$\frac{f(x)-f(a)}{x-a} = \frac{\sqrt{1-2x} - \sqrt{1-2a}}{x-a}$$

Rationalize the second difference quotient

Rationalize the second difference quotient by multiplying both numerator and denominator by the conjugate of the numerator: $$\frac{\sqrt{1-2x} - \sqrt{1-2a}}{x-a} \cdot \frac{\sqrt{1-2x} + \sqrt{1-2a}}{\sqrt{1-2x} + \sqrt{1-2a}} = \frac{(1-2x)-(1-2a)}{(x-a)(\sqrt{1-2x} + \sqrt{1-2a})}$$

Simplify the second difference quotient

Simplify the second difference quotient: $$\frac{2a-2x}{(x-a)(\sqrt{1-2x} + \sqrt{1-2a})} = \frac{-2(x-a)}{(x-a)(\sqrt{1-2x} + \sqrt{1-2a})} = \frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}$$ The simplified difference quotients are: $$\frac{f(x+h)-f(x)}{h} = \frac{-2}{\sqrt{1-2x-2h} + \sqrt{1-2x}}$$ $$\frac{f(x)-f(a)}{x-a} = \frac{-2}{\sqrt{1-2x} + \sqrt{1-2a}}$$