Odd Even Functions E4A5F9

1. **Define odd and even functions with examples:** - An **even function** satisfies $$f(-x) = f(x)$$ for all $$x$$ in its domain. - An **odd function** satisfies $$f(-x) = -f(x)$$ for all $$x$$ in its domain. **Examples:** - Even function: $$f(x) = x^2$$ because $$(-x)^2 = x^2$$. - Odd function: $$f(x) = x^3$$ because $$(-x)^3 = -x^3$$. 2. **Calculate $$f(2) + f(-2)$$ for $$f(x) = x^2 + 6x + 10$$:** - First, find $$f(2)$$: $$f(2) = 2^2 + 6(2) + 10 = 4 + 12 + 10 = 26$$. - Next, find $$f(-2)$$: $$f(-2) = (-2)^2 + 6(-2) + 10 = 4 - 12 + 10 = 2$$. - Sum: $$f(2) + f(-2) = 26 + 2 = 28$$. 3. **Differentiate $$y = e^x + \\log x + 1089$$ with respect to $$x$$:** - Recall: $$\\frac{d}{dx} e^x = e^x$$ $$\\frac{d}{dx} \\log x = \\frac{1}{x}$$ $$\\frac{d}{dx} 1089 = 0$$ - Therefore: $$\\frac{dy}{dx} = e^x + \\frac{1}{x}$$. 4. **Find the range and coefficient of range for the data:** $$-40, 52, 47, 28, 45, 36, 47, 50$$ - Range is $$\text{max} - \text{min}$$: $$\text{max} = 52$$ $$\text{min} = -40$$ $$\text{Range} = 52 - (-40) = 92$$. - Coefficient of range is: $$\\frac{\text{Range}}{\text{max} + \text{min}} = \\frac{92}{52 + (-40)} = \\frac{92}{12} = 7.67$$. 5. **Compare consistency of two sets of observations:** Given: - Set I: mean $$x = 82.5$$, standard deviation $$s = 48.75$$ - Set II: mean $$x = 73$$, standard deviation $$s = 8.35$$ - Consistency is measured by coefficient of variation (CV): $$CV = \\frac{s}{mean}$$ - Calculate CV for each: $$CV_1 = \\frac{48.75}{82.5} = 0.59$$ $$CV_2 = \\frac{8.35}{73} = 0.11$$ - Since $$CV_2 < CV_1$$, **Set II is more consistent**. **Final answers:** - Odd and even functions defined with examples. - $$f(2) + f(-2) = 28$$. - $$\\frac{dy}{dx} = e^x + \\frac{1}{x}$$. - Range = $$92$$, Coefficient of range = $$7.67$$. - Set II is more consistent.