1. Жұп функцияны табыңыз. A) \( y=-x^{2}|x|+x^{5} \) B) \( y=x^{2}|x|+x^{2} \) C) \( y=x^{3}-x|x| \) D) \( y=x^{4}|x|+x^{3} \) 2. \( f(x)=\cos \left(\frac{5 x}{2}-\frac{\pi}{6}\right)+1 \) функциясының ең кіші оң периодын табыңыз. A) \( \frac{3 \pi}{2} \) B) \( \frac{3 \pi}{5} \) C) \( \frac{3 \pi}{7} \) D) \( \frac{4 \pi}{5} \) 3. Жұп функцияны көрсетіңіз. A) \( y=3 x^{2}-x^{4} \) B) \( y=6 x^{5}-2 x^{4}+8 \) C) \( y=\frac{4}{x} \) D) \( y=\frac{x+4}{x-4} \) 4. \( y=\left|\sin \frac{x}{3}\right|-11 \) функциясының ең кіші периодын табыңыз. A) \( 2 \pi \) B) \( \frac{2}{3} \pi \) C) \( 6 \pi \) D) \( 3 \pi \) 5. \( y=\sqrt[5]{x+4}+\sqrt[5]{x-4} \) функциясының жұп немесе тақ екенін анықтаңыз. A) Жпп B) Тақ C) Жұп та, тақта емес D) Анықтау мумкін емес 6. \( y=\sqrt[5]{x+14}+\sqrt[7]{-x+14} \) функциясының жұп немесе тақ екенін анықтаңыз. A) Жұп B) Тақ C) Жұп та, тақта емес D) Анықтау мумкін емес

Function by following steps: - step0: Determine if even, odd or neither: \(y=3x^{2}-x^{4}\) - step1: Use the appropriate notation: \(f\left(x\right)=3x^{2}-x^{4}\) - step2: Use substitution: \(f\left(-x\right)=3\left(-x\right)^{2}-\left(-x\right)^{4}\) - step3: Simplify: \(f\left(-x\right)=3x^{2}-x^{4}\) - step4: Use substitution: \(f\left(-x\right)=f\left(x\right)\) - step5: The function is an even function: \(\textrm{Even}\) Determine whether the function \( y=x^{3}-x|x| \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=x^{3}-x\left|x\right|\) - step1: Use the appropriate notation: \(f\left(x\right)=x^{3}-x\left|x\right|\) - step2: Use substitution: \(f\left(-x\right)=\left(-x\right)^{3}-\left(-x\left|-x\right|\right)\) - step3: Simplify: \(f\left(-x\right)=-x^{3}+x\left|x\right|\) - step4: Use substitution: \(f\left(-x\right)=-f\left(x\right)\) - step5: The function is an odd function: \(\textrm{Odd}\) Determine whether the function \( y=x^{4}|x|+x^{3} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=x^{4}\left|x\right|+x^{3}\) - step1: Use the appropriate notation: \(f\left(x\right)=x^{4}\left|x\right|+x^{3}\) - step2: Use substitution: \(f\left(-x\right)=\left(-x\right)^{4}\left|-x\right|+\left(-x\right)^{3}\) - step3: Simplify: \(f\left(-x\right)=x^{4}\left|x\right|-x^{3}\) - step4: The function is neither even nor odd: \(\textrm{Neither even nor odd}\) Analyze the periodicity of the function \( y=\frac{1}{3}\sin(x) - 11 \) Function by following steps: - step0: Determine the period: \(y=\frac{1}{3}\sin\left(x\right)-11\) - step1: The period of the function is \(\frac{2\pi }{\left|1\right|}:\) \(\frac{2\pi }{\left|1\right|}\) - step2: Calculate: \(\frac{2\pi }{1}\) - step3: Divide the terms: \(2\pi \) Determine whether the function \( y=-x^{2}|x|+x^{5} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=-x^{2}\left|x\right|+x^{5}\) - step1: Use the appropriate notation: \(f\left(x\right)=-x^{2}\left|x\right|+x^{5}\) - step2: Use substitution: \(f\left(-x\right)=-\left(-x\right)^{2}\left|-x\right|+\left(-x\right)^{5}\) - step3: Simplify: \(f\left(-x\right)=-x^{2}\left|x\right|-x^{5}\) - step4: The function is neither even nor odd: \(\textrm{Neither even nor odd}\) Determine whether the function \( y=x^{2}|x|+x^{2} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=x^{2}\left|x\right|+x^{2}\) - step1: Use the appropriate notation: \(f\left(x\right)=x^{2}\left|x\right|+x^{2}\) - step2: Use substitution: \(f\left(-x\right)=\left(-x\right)^{2}\left|-x\right|+\left(-x\right)^{2}\) - step3: Simplify: \(f\left(-x\right)=x^{2}\left|x\right|+x^{2}\) - step4: Use substitution: \(f\left(-x\right)=f\left(x\right)\) - step5: The function is an even function: \(\textrm{Even}\) Determine whether the function \( y=\frac{4}{x} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=\frac{4}{x}\) - step1: Use the appropriate notation: \(f\left(x\right)=\frac{4}{x}\) - step2: Use substitution: \(f\left(-x\right)=\frac{4}{-x}\) - step3: Rewrite the fraction: \(f\left(-x\right)=-\frac{4}{x}\) - step4: Use substitution: \(f\left(-x\right)=-f\left(x\right)\) - step5: The function is an odd function: \(\textrm{Odd}\) Determine whether the function \( y=6 x^{5}-2 x^{4}+8 \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=6x^{5}-2x^{4}+8\) - step1: Use the appropriate notation: \(f\left(x\right)=6x^{5}-2x^{4}+8\) - step2: Use substitution: \(f\left(-x\right)=6\left(-x\right)^{5}-2\left(-x\right)^{4}+8\) - step3: Simplify: \(f\left(-x\right)=-6x^{5}-2x^{4}+8\) - step4: The function is neither even nor odd: \(\textrm{Neither even nor odd}\) Determine whether the function \( y=\frac{x+4}{x-4} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=\frac{x+4}{x-4}\) - step1: The function is neither even nor odd: \(\textrm{Neither even nor odd}\) Determine whether the function \( y=\sqrt[5]{x+4}+\sqrt[5]{x-4} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(y=\sqrt[5]{x+4}+\sqrt[5]{x-4}\) - step1: Determine if even, odd or neither: \(\textrm{Odd}\) Давайте разберем каждую задачу по порядку. ### 1. Найдите четную функцию. - A) \( y=-x^{2}|x|+x^{5} \) - **Не четная и не нечетная** - B) \( y=x^{2}|x|+x^{2} \) - **Четная** - C) \( y=x^{3}-x|x| \) - **Нечетная** - D) \( y=x^{4}|x|+x^{3} \) - **Не четная и не нечетная** **Ответ:** B) \( y=x^{2}|x|+x^{2} \) ### 2. Найдите наименьший положительный период функции \( f(x)=\cos \left(\frac{5 x}{2}-\frac{\pi}{6}\right)+1 \). Период функции косинуса определяется как \( \frac{2\pi}{k} \), где \( k \) - коэффициент перед \( x \). В данном случае \( k = \frac{5}{2} \), следовательно, период будет: \[ P = \frac{2\pi}{\frac{5}{2}} = \frac{4\pi}{5} \] **Ответ:** D) \( \frac{4\pi}{5} \) ### 3. Укажите четную функцию. - A) \( y=3 x^{2}-x^{4} \) - **Четная** - B) \( y=6 x^{5}-2 x^{4}+8 \) - **Не четная и не нечетная** - C) \( y=\frac{4}{x} \) - **Не четная и не нечетная** - D) \( y=\frac{x+4}{x-4} \) - **Не четная и не нечетная** **Ответ:** A) \( y=3 x^{2}-x^{4} \) ### 4. Найдите наименьший период функции \( y=\left|\sin \frac{x}{3}\right|-11 \). Период функции синуса равен \( 2\pi \). В данном случае период будет: \[ P = 2\pi \cdot 3 = 6\pi \] **Ответ:** C) \( 6\pi \) ### 5. Определите, является ли функция \( y=\sqrt[5]{x+4}+\sqrt[5]{x-4} \) четной или нечетной. Функция является нечетной, так как: \[ f(-x) = \sqrt[5]{-x+4}+\sqrt[5]{-x-4} = -f(x) \] **Ответ:** B) Нечетная ### 6. Определите, является ли функция \( y=\sqrt[5]{x+14}+\sqrt[7]{-x+14} \) четной или нечетной. Функция является нечетной, так как: \[ f(-x) = \sqrt[5]{-x+14}+\sqrt[7]{x+14} = -f(x) \] **Ответ:** B) Нечетная ### Итоговые ответы: 1. B 2. D 3. A 4. C 5. B 6. B