共50条信息
已知函数\(y=f(x)\)满足:对任意\(a\),\(b∈R\),\(a\neq b\),都有\(af(a)+bf(b) > af(b)+bf(a)\).
\((1)\)试证明:\(f(x)\)为\(R\)上的单调增函数.
\((2)\)若\(x\),\(y\)为正实数且\(\dfrac{4}{x} +\dfrac{9}{y} =4\),比较\(f(x+y)\)与\(f(6)\)的大小.
定义在\(R\)上的奇函数\(f\left( x \right)\)满足条件\(f\left( 1+x \right)=f\left( 1-x \right)\),当\(x∈\left[0,1\right] \)时,\(f\left( x \right)=x\),若函数\(g\left( x \right)=\left| f\left( x \right) \right|-a{{e}^{-\left| x \right|}}\)在区间\(\left[-2018,2018\right] \)上有\(4032\)个零点,则实数\(a\)的取值范围是( )
已知函数\(y{=}f(x)\)的定义域\({[-}8{,}1{]}\),则函数\(g(x){=}\dfrac{f(2x{+}1)}{x{+}2}\)的定义域是\(({ })\)
函数\(f(x)\)的定义域为\(D=\{x|x\neq 0\}\),且满足对于任意\(x_{1}\),\(x_{2}∈D\),有\(f(x_{1}·x_{2})=f(x_{1})+f(x_{2}).\)
\((1)\)求\(f(1)\)的值;
\((2)\)判断\(f(x)\)的奇偶性并证明你的结论;
\((3)\)如果\(f(4)=1\),\(f(x-1) < 2\),且\(f(x)\)在\((0,+∞)\)上是增函数,求\(x\)的取值范围.
若定义在\(R\)上的函数\(f\left(x\right) \)满足:对任意的\({x}_{1},{x}_{2}∈R \)有\(f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)+2, \)则下列说法一定正确的是( )
若函数\(f(x)\)的定义域为\((-2,2)\),则函数\(g(x)=f(x-1)+f(3-2x)\)的定义域为________.
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