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SAT考试数学考点之函数精讲

2016-12-08 11:24
摘要:SAT数学部分所考查的知识点,包括基础代数、数据分析&实际问题、中级代数三个部分,对于中国学生而言,这些知识点难度不大,但部分知识点考查的方式与侧重点和国内考试有些差异,需要注意。

SAT数学部分所考查的知识点,包括基础代数、数据分析&实际问题、中级代数三个部分,对于中国学生而言,这些知识点难度不大,但部分知识点考查的方式与侧重点和国内考试有些差异,需要注意。


以下就是100留学为大家带来的SAT考试数学考点之函数精讲。希望对各位的备考有所帮助


1.1函数的基本概念

函数的定义

A functionfis a correspondence that associateswith each elementaof a set calledthe domain (定义域) oneandonly one elementof a set called the range(值域).

函数的零点

The zeros of a functionfare the values ofxfor whichf(x)=0; theyare thex-intercepts of the graph ofy=f(x).


例题1. Gracekicks a soccer ball into the air, where the height of the ball follows thefunction h(t) = 8t – t2. After how many seconds does the ball returnto the ground?

(A) 0

(B) 4

(C) 6

(D) 8

Solution:

球落到地面的数学含义是h(t)=0.

h(t) = 8t – t2=0

t(8-t)=0

t=0 OR t=8

D



4adc30f64757e8b79d85c56.png

例题2. Whatis a possible equation for the linear function above?

(A)y = –10/7x– 3

(B)y =–10/7x + 3

(C)y =10/7x– 3

(D)y =10/7x+ 3

Solution:

图像中最重要的两个特征:

1)直线的斜率为负值;

2)直线与X轴和Y轴的交点均为正值。

由此可以得到正确选项,B

反函数

Iffisone-to-one with domain X and range Y,then there is a function, with domain Y and range X, such that

函数的复合Combinationof functions

The composition offwithg, written asf(g(x))isthe function obtained by replacingxwherever it occurs inf(x)byg(x). We also write forf(g(x)). The domain of is the set of allxin the domain ofgfor whichg(x)is inthe domain off.


例题3. Iff(x) = 2x and g(x) = 5x + 1, what is g (f(c))?

(A) 2c

(B) 10c + 1

(C) 10c

(D) 20c + 2

Solution

g (f(x))=5(2x)+1=10x+1             

B

例题4. If f (x) = 8x + 1 and g (x) = 3x – 1, whatis the value of?

Solution


答案:

奇偶函数Oddand even functions

ODD functions:

EVEN functions:


绝对值函数absolute-valuefunctions

绝对值函数以及含有绝对值的不等式是考查的一个重点,其包含了多重意义。例如:

以上不等式的解表示的是数轴上到3的距离小于等于2的所有点的集合。


b5a9d7835357e8b7ab66f93.png

例题5. Whatinequality is represented by the graph above?

(A)y ≥ |x –2|

(B)y ≥ |x +2|

(C)y ≤ |x –2|

(D)y ≤ |x +2|

Solution:

含有绝对值的不等式,其解往往是一个范围,而不是单个数值。图中折线的函数是,在折线上方的就是。

答案:A


例题6. If|3x – 1| ≤ 2x, where x > 0, what is a possible value of x?

Solution

1



所以:

(2)



所以:

综合(1)(2),x的范围是。

注意这题的要求是给出一个可能的解就行,以上是完整的解题过程,但在考试的时候只要计算其中一种情况,得出一个合适的解就行了,不用全解,但要注意验算。


例题7. If x≤ 9 and x ≥ 1, which of the following statements are true?

I. – 1 ≤ x ≤ 9

II. 1 ≤ x ≤ 9

III. |x – 5| ≤ 4

(A) I only

(B) II and III

(C) I and II

(D) I, II and III

Solution


取值范围有小及大可以,由大及小不行。

答案:D



64e485959957e8b7bbc74dc.png

例题8. If(x, y) is the solution to the system of equations graphed above, what is thevalue of x in terms of y?

(A)y

(B)2/3y

(C)1/3y

(D)

Solution:

方程组的解即为图中两条直线的交点,所以

答案:B


例题9. Leois manufacturing rulers. If the ruler differs from the expected length by 1 mm,he needs to throw it away. What absolute value inequality represents the rulersthat Leo does NOT throw away?

(A)|x – 1|≤ 0.01

(B)|x – 1|≤ 0.001

(C)|x – 1|≥ 0.01

(D)|x – 1|≥ 0.001

Solution

要求误差在0.001m之内,所以是|x – 1| ≤ 0.001

答案:B



取整函数thegreatest-integer function

函数图像

常见的几种图像变换:

Supposedefines any function:

1.translates k units vertically (up if k>0; down if k<0);

2.translates h units horizontally (right if h>0; left if h<0);

3.stretches (shrinks) f(x) vertically by a factor of |a| if |a|>1 (|a|<1);

4.stretches (shrinks) f(x) horizontally by a factor of if |a|>1 (|a|<1);

5. reflectsabout the x-axis. (the reflection is vertical.);

6. reflectsabout the y-axis. (the reflection is horizontal.);


af1c66313457e8b7e90e0a9.png

例题10. Thefunctionf(x) is graphed above. Ifg(x) =f(x) – 1,which of thefollowing statements is TRUE?

(A) g(x) is greater than or equal to zero.

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