Please answer a few questions about the function of the first year of high school

1.Let f(x)=ax+b, then f(f(x))=a(ax+b)+b=2x-1, so a 2=2, 2ab = -1 (comparison coefficient).

So a = root number 2, b = - root number 2 4 or a = - root number 2, b = root number 2 4

2.(1) Since f(x) defines the domain as [1,2), it requires 1<=x 2<2 for f(x 2), and solving this inequality can give the domain defined as (-root2,-1] and [1,root2).

2) Similar to (1), since 0<=x+1<=1, so -1<=x<=1 so f(x) defines the domain as [

3.Because the volume is 8000 cubic meters and the depth is 6 meters, the bottom area is 8000 6 = 4000 3 square meters.

If one side of the ground is x, then the other side must be 4000 (3x), so you can use the box area formula and notice that there are four sides, so there is.

y=2a*4000/3+2*a*x*6+2*a*6*4000/(3x)

So y=8000a 3+12a*x+16000a x

By the mean inequality, 12ax + 16000a x > = 2 * root number (12ax * 16000a x) = 160a * root number 30

There is a minimum value of y, and it can be seen that there is no maximum value of y.

4.Since f(x) is an odd function, f(-x) = -f(x).

So when x is less than 0, -x > 0

So, f(-x) = (-x) (the cube root of 1+3).

So, f(-x) = (-x) (the cube root of 1+3).

5.Because f is an odd function.

So the image of f is symmetrical with respect to the origin.

Therefore, the origin is either undefined, or f(0)=0

If we know that f(0)=0 can be determined, then there is the first equation for m,n.

and by f(-x)=-f(x), there is a second equation for m,n.

Combining the above two equations, we can solve m,n (the process is slightly first, and I will go out to class soon).

6.Since the range of f(x)=(ax+1) (x 2+c) is [-1,5].

So the inequality -1<=(ax+1) (x 2+c)<=5 holds for any real number x.

That is, there are groups of inequalities.

x^2+ax+c+1)/(x^2+c)>=0

5x^2+ax+1-5c)/(x^2+c)<=0

That is, (x 2+ax+c+1)>=0, (x 2+c)>0 or (x 2+ax+c+1)<=0, x 2+c)<0

and (-5x 2+ax+1-5c)(x 2+c)<=0.

Then you can move on to the discussion.

In the first question, replace 2x-1 with the x in 2x-1, i.e., 2(2x-1)-1....f(x) is obtained as 4x-3

f(x)=ax 2-2ax+3-b with an axis of symmetry x=1

a>0, the image opening is directed upward, the function f(x) is increasing in the pure [1,3] of the burning limb, and the hunger is f(1)=-a+3-b=2, f(3)=3a+3-b=5, and the solution is a=3 4, b=1 4

1. If f(x) is an even function, then there is: f(-x)=f(x).

f(x)=x^2+|x-a|+1...1)

f(-x)=x^2+|-x-a|+1...2)

Formula (1) = Formula (2), get.

x-a|=|x+a|So, a=0

2. Suppose there is a real number a, such that the function f(x) is an odd function, then there is:

f(-x)=-f(x)

f(-x)=x^2+|-x-a|+1...3)

f(x)=-(x^2+|x-a|+1)..4)

Formula (1) = Formula (2), get.

2x^2+|x+a|+|x-a|+2=0...5)

Because :x belongs to r, 2x 2>=0,|x+a|>=0，|x-a|>=0, i.e. .

2x^2+|x+a|+|x-a|>=0, obviously Eq. (5) does not hold.

Therefore, no matter if a takes any real function f(x), it cannot be an odd function.

1 is an even function, then f(x)=f(-x) x 2+|x-a|+1=x^2+|-x-a|+1 So|-x-a|=|x-a|，a=0

2 Because x has a definition on 0, it must be satisfied that f(x) is an odd function when f(0)=0, and it is obvious that f(0) is greater than or equal to 1, so it cannot be an odd function.

1.Because the function is even and increases in a positive interval.

So it decreases in the negative range.

and because f(-3)=0

So f(3)=0

According to the increase or decrease of x<-3 or x>3, f(x)>0g(x)=x2+3x+2a

1) Because both a and b contain 2

So f(2)=0 g(2)=0

a=-5f(x)=2x 2-5x+2 is 1 2 and 2g(x)=x 2+3x-10 is 2 and -5a=b=(2)u=

cua=cub=

cua u cub=

3),,empty.

3.Because of symmetry.

So when x 1 image over (1,1)(2,0).

From the parabolic vertex (0,2), we get ax 2+bx+c, where b=0, c=2, i.e., ax 2+2 and (1,1)a=-1

i.e. -x 2+2 (-1 x 1).

f(x)= x+2(x≤-1)

x^2+2(-1≤x≤1)

x+2(x≥1)

Monotonous: (-0] increase [0, + minus.

1.Range(-3,0)u(0,3).

2.Since anb=, 2 a and 2 b

2*2^2+2a+2=0

2^2+3*2+2a=0

The solution yields a = -5

Then a= b=

So cua= cub=

cua)u(cub)=

There are eight subsets in total. ，3.Less than -1 is a primary curve.

Substituting (-2,0)(-1,1) to solve the equation is y=x+2, and since the symmetry is greater than 1, the equation is y=-x+2

When between -1 and 1.

Use the vertex of the parabola to set y=ax 2+2

Substituting (-1,1) points.

The solution is a=-1

So. f(x)=-x+2(x≥1)

x^2+2(-1x+2(x≤-1)

Image omitted. Monotonic interval (-infinity, 0) (0, +infinity) The former increases and the latter decreases.

3^a=5^b=a

a=log3(a)

b=log5(a)

1/log3(a)+1/log5(a)=2log3(a)+log5(a)=2log3(a)log5(a)lga/lg3+lga/lg5=2lga/log3*lga/lg5lga(lg5+lg3)=2(lga)^2lga(lg3+lg5-2lga)=0

LGA = 0 or 2LGA = LG15

a = 1 or a = 15

When a=1, a=b=0, it does not meet the question setting, and it is rounded, so a= 15

First calculate b a=2b-1, according to the power of 3 a = 5 to the power b, get b a=, the simultaneous equation, get b, bring in a = 5 to the power b, get the answer.

1、f(x)=(2x+2a-2a+1)/(x+a)=(2x+2a)/(x+a)+(1-2a)/(x+a)=2+(1-2a)/(x+a)

It can be seen from the knowledge of the inverse proportional function.

x+a<0 or x+a>0, it is a monotonic function.

i.e. x<-a or x>-a

Here is x>-1 monotonous.

So it should be included in x>-a

So -1 -a

a 12, odd function, x>0 increments then x<0 also increases.

Odd function then f(-x) = -f(x).

So [f(x)-f(-x)] x<0

i.e. [f(x)+f(x)] x<0

2f(x)/x<0

When x<0, f(x) >0

Because it is an odd function, f(-1)-f(1)=0 is f(x)>f(-1), and the function is increased.

At 10, f(x)<0=f(1).

The function 00 is added, so f(-x) applies to f(x)=x(the cube root of 1+x) f(-x)=-x(the cube root of 1-x).

Odd function then f(-x) = -f(x).

So (-0) on f(x)=x(the cube root of 1-x).

The axis of symmetry of the function is changed to x=a 2

When A2 0 and X=0, Y is taken to the maximum.

So -a 4+1 2=2

a=-6 when a2 0,1 and x=a2, y is taken to the maximum.

So-(a Air Fighter 2) 2+a*a 2-a 4+1 2=2a=3 (rounded) or a=-2 (rounded).

When A2 1, X=1, Y is taken to the maximum.

So -1+a-a 4+1 2=2

a=3 10 (rounded to fight).

In summary, the complaint a=-6

Since it is an odd function, f(0)=0 calculates b=0 f(1 2) and brings in the function a=4 5

Bringing the values of a and b into the equation gives us the analytic formula.

The analytic formula deriving first can determine its monotonicity.

Find b, f(1 2) find a

2.Derivative judgment is sufficient.

Since the function is odd, f(x)=-f(-x), so that you can list a formula, the contrast coefficient can be found b=0, and then the value of f(1 2) can be found a.

The proof of monotonicity should be a derivative.

intersects with the x-axis a(-2,0) and b(4,0), then y=a[x-(-2)](x-4).

a(x²-2x-8)

a(x²-2x+1-9)

a[(x-1)²-9]

The maximum value is -9a=9

a=-1 opening downwards, so there is a maximum value for conformance.

So y=-x +2x+8