高中课件 函数.docx

Lesson 2-Function PropertiesMonotonic function:In calculus, a function  defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. Figure 1It is called monotonically increasing (also increasing or non-decreasing), if for all  and  such that  one has , so preserves the order (see Figure 1). Figure 2Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever , then , so it reverses the order (see Figure 2).Important conclusions:1、若函数y=f(x) 和y=g(x) 在公共区间A内都是增（减）函数，则函数y=f(x)+g(x)在A是增（减）函数。2、若两个正值函数（所有值为正的函数）y=f(x) 和y=g(x)在公共区间A内都是增（减）函数，则函数y=f(x)g(x)在区间A内是增（减）函数。3、若两个负值函数（所有值为负的函数）y=f(x) 和y=g(x)在公共区间A内都是增（减）函数，则函数y=f(x)g(x)在区间A内是减（增）函数。4、设函数y=f(u)和u=g(x)在公共区间A内都是单调函数，那么函数y=fg(x)在A内的也是单调函数；并且，若y=f(u)和u=g(x)的单调性相同（反），则y=fg(x)是增（减）函数，这个性质可以按下面的两种方式去记忆。i) 同增异减：y=f(u)与u=g(x)增减性相同（相反），函数y=fg(x)是增（减）函数。ii)这里用“+”代表“递增”，用“-”代表“递减”，随后复合函数的单调性可按下表记忆。内层函数:u=g(x)+-外层函数:y=f(u)+-+-复合函数:y= fg(x)+-+Practice:1. 判断函数f(x)=-x3+a（a R)在R上的单调性。2、已知函数f(x)与g(x)在R上都是增函数，判断fg(x)在R上也是增函数。3、判断函数y=x2+1/x在（- ，0）上的单调性。Even and odd functions:Even functions:Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:F(x)=F(-x)Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.Odd functions:Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:F(x)=-F(-x)Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.Uniqueness properties: If a function is even and odd, it is equal to 0 everywhere it is defined.Properties involving addition and subtraction:The sum of two even functions is even, and any constant multiple of an even function is even.The sum of two odd functions is odd, and any constant multiple of an odd function is odd.The difference between two odd functions is odd.The difference between two even functions is even.The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain.Properties involving multiplication and division:The product of two even functions is an even function.The product of two odd functions is an even function.The product of an even function and an odd function is an odd function.The quotient of two even functions is an even function.The quotient of two odd functions is an even function.The quotient of an even function and an odd function is an odd function.Properties involving composition:The composition of two even functions is even.The composition of two odd functions is odd.The composition of an even function and an odd function is even.The composition of either an odd or an even function with an even function is even (but not vice versa).Practice:1、证明反比例函数f(x)= k/x是奇函数。2、判断下列函数的奇偶性（1）y=-x2(x3)（2）f(x)= 3x-4x-2（3）y= x+1x（4）f(x)=x4+1x2+13、判断f(x)= -x2+x(x>0) 的奇偶性 x2=x(x0)4、已知函数f(x)对一切x,yR都有f(x+y)=f(x)+f(y)。（1）求证f(x)是奇函数（2）设f(-3)=a，用a表示f(12).Inverse function:Definition:Let f be a function whose domain is the set X, and whose range is the set Y.Then f is invertible if there exists a function g with domain Y and range X, with the property:If f is invertible, the function g is unique, which means that there is exactly one function g satisfying this property (no more, no less). That function g is then called the inverse of f, and is usually denoted as f1.Important conclusions:1、函数的定义域就是它的反函数的值域，函数的值域就是它的反函数的定义域。2、在同一坐标系内，函数与其反函数的图像关于y=x对称。3、如果函数y=f(x)的图像关于直线y=x对称，那么它存在反函数，并且反函数就是它本身。4、若函数y=f(x)是A上的增（减）函数，则其反函数y=f-1(x)是C上的增（减）函数。（A,C是f(x)的定义域，值域）。5、奇函数若有反函数，则反函数仍是奇函数，偶函数不存在反函数。Practice:1、求函数y=x 的反函数2、已知y=12x +m 和y=nx-13互为反函数，求m与n的值3、求函数y=x2+2(x0)的反函数的定义域4、已知函数y=x-52x+m的图像关于直线y=x对称，求实数m的值5、设点（1，2）既在y=ax+b 的图像上又在其反函数的图像上，求a与b的值6、已知f(x)=x2-1(x2)，求f-1(4)