共50条信息
\((1)\)若函数\(y{=}2^{{-}{|}x{+}3}{|}\)在\(({-∞}{,}t)\)上是单调增函数,则实数\(t\)的取值范围为______ .
\((2)\)已知\(a{ > }0\),则\(\dfrac{(a{+}1)^{2}}{a}\)的最小值为______.
\((3)\)某班共\(50\)人,其中\(21\)人喜爱篮球运动,\(18\)人喜爱乒乓球运动,\(20\)人对这两项运动都不喜爱,则喜爱篮球运动但不喜爱乒乓球运动的人数为______ .
\((4)\)若对于任意正数\(x{,}y\),都有\(f({xy}){=}f(x){+}f(y)\),且\(f(8){=-}3\),则\(f(a){=}\dfrac{1}{2}\)时,正数\(a{=}\) ______ .
设函数\(y=f\left(x\right) \)是定义在\(R\)上的函数,并且满足下面三个条件:\(①\)对任意正数\(x\),\(y\),都有\(f\left(xy\right)=f\left(x\right)+f\left(y\right) \);\(②\)当\(x > 1 \)时,\(f\left(x\right) < 0 \);\(③f\left(3\right)=-1 \).
\((2)\)证明\(f\left(x\right) \)在\(\left(0,+∞\right) \)上是减函数;
已知函数\(f(x)= \dfrac{a}{x}+x\ln x\),\(g(x)=x^{3}-x^{2}-5\),若对任意的\(x_{1}\),\(x_{2}∈\left[ \left. \dfrac{1}{2},2 \right. \right]\),都有\(f(x_{1})-g(x_{2})\geqslant 2\)成立,则\(a\)的取值范围是\((\) \()\)
设函数\(f(x)(x∈R)\)满足\(f(x+π)=f(x)+\sin x\),当\(0\leqslant x < π\)时,\(f(x)=0\),则\(f(\dfrac{23\pi }{6})=(\) \()\)
若\(f(x)\) 满足对任意的实数\(a\),\(b\)都有\(f(a+b)=f(a)f(b)\) 且\(f(1)=2\) ,则\( \dfrac{f(2)}{f(1)}+ \dfrac{f(4)}{f(3)}+ \dfrac{f(6)}{f(5)}+…+ \dfrac{f(2 016)}{f(2 015)}=\)( )
定义在\(R\)上的函数\(f(x),f(0)\ne 0,f(1)=2\),当\(x > 0,f(x) > 1\),且对任意\(a,b\in R\),有\(f(a+b)=f(a)\cdot f(b)\) .
\((1)\)求证:对任意\(x\in R\),都有\(f(x) > 0\);
\((2)\)判断\(f(x)\)在\(R\)上的单调性,并用定义证明;
\((3)\)求不等式\(f(3-2x) > 4\)的解集.
\(18.\)已知二次函数\(f(x)=ax^{2}+bx+c\)满足\(f(0)=1\),对任意\(x∈R\),都有\(1-x\leqslant f(x)\),且\(f(x)=f(1-x)\).
\((\)Ⅰ\()\)求函数\(f(x)\)的解析式;
\((\)Ⅱ\()\)若\(∃x∈[-2,2]\),使方程\(f(x)+2x=f(m)\)成立,求实数\(m\)的取值范围.
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