Math problems in the first year of high school are taught by people who know

1.Let f(x)=kx+b

then f(f(x)))=k(kx+b)+b

k^2x+kb+b=4x+4

The coefficients correspond to one to one.

k^2=4kb+b=4

Solve k= or k=-2 b=-4

So y=2x+4 3 or y=-2x-4

2.Make the same use of the methods above.

y=ax^2+bx+c

f(x+4)=a(x+4)^2+b(x+4)+c=ax^2+(8a+b)x+16a+4b+c

f(x-1)=a(x-1)^2+b(x-1)+c=ax^2+(b-2a)x+a-b+c

f(x+4)+f(x-1)=2ax^2+(6a+2b)x+17a+3b+2c =x^2-2x

2a = 1, and 6a + 2b = -2, and 17a + 3b + 2c = 0

a=1/2b=-5/2

c=-1/2

f(x)=(x^2)/2-5x/2-1/2

f(m)=m^2-2008m

f(n)=n^2-2008n

Subtract the two formulas f(m)=f(n).

m^2-n^2)-2008(m-n)=(m-n)(m+n-2008)=0

m is not equal to n, then m+n=2008

f(m+n)=f(2008)=2008^2-2008*2008=0

It's not a complicated question.

For the first problem, we can use the undetermined coefficient method, which is a very useful method, because once and twice, we already know the general form of the analytic formula, 1) we can let f(x)=ax+b

Then substitution is sufficient, that is, a(ax+b)+b=4x+4.

From this we can find a=2, b=4 3Or a=-2, b=-4

2) F(x)=ax*x+bx+c

Then there is, a(x+4)(x+4)+b(x+4)+c+a(x-1)(x-1)+b(x-1)+c=x*x-2x

After tidying up. The corresponding coefficients are equal to find the value of ABC, and thus the analytic formula.

2.This question is actually an image of a quadratic function.

In particular, the axis of symmetry is examined. It is easy to know that the axis of symmetry is x=1004Since the function values of mn are equal, it means that mn is symmetrical with respect to the axis of symmetry.

i.e. m+n=2008So the following things are easy to do. Just substitute the evaluation.

Obviously, the result is 0

1. Vector od=1 2 vector oc=1 2 (vector oa + vector ob) = 1 2 (vector a + vector b).

2. Make auxiliary lines: extend AD to E so that AD=DE connects BE and CE, then the quadrilateral ABCE is a parallelogram (the parallelogram diagonals are bisected with each other), and then the practice is the same as 1: vector AD=1 2 Vector AE1 2 (vector AB+vector AC)=1 2(vector A + vector B).

Vector od=12 (vector oa + vector ob).

vector ad=12 (vector ab + vector ac).

1. tan(60°)=tan(20°+tan40°)=(tan20°+tan40°) (1-tan20° tan40°), get tan20°+tan40°=tan60° (1-tan20° tan40°) = root number 3 - root number 3 tan20° tan40°, substitute the original merger.

2、tan(a+b)=tan225°=1