Because f(x) is an even function and g(x) is a odd function, the above formula is -g(t)=f(t+ 1), that is, g(x)=-f(x+ 1).

So f(x- 1)=-f(x+ 1), that is, f(x- 1)+f(x+ 1)=0.

When x=20 14, f (2013)+f (2015) = 0.

Is F(x) = f (-x) because of F(x) even function?

F(x- 1) odd function, so f (x-1) =-f (-x-1) =-f [-(x+1)] =-f (x+/kloc-0.

That is, f (x) =-f (x+2) = f (x+4) =-f (x+6) = f ... = (-1) (n/2) f (x+n).

Whenever the difference between values is 2, the function value changes sign once.

The difference between 0.5 and 8.5 is four twos, so the sign is changed to four times.

So f(8.5)=f(0.5)=9.

The image of y = f (x) is obtained by moving the image of y=f(x+2) to the right by two units.

And y=f(x+2) is an even function, that is, the image of y=f(x+2) is symmetrical about y,

So the image of y=f(x) is symmetrical about x=2, f(4)=f (0), f(5.5)=f (- 1.5).

So f(5.5) is less than f(- 1) is less than f(4).

F(x) is a continuous even function, when x >; 0, f(x) is a monotone function.

So when x

For example, if f(a)=a, then only f (-a) = f (a) = a.

Let f(x)=f((x+3)/(x+4))

f(x)=f(-x)

So we can get two equations.

X=(x+3)/(x+4), that is, x 2+3x-3 = 0.

-x=(x+3)/(x+4), that is, x 2+5x+3 = 0.

Through Vieta's theorem? The sum of all x is (-3)+(-5)=-8?

Let f (x) = f (x)+x.

∫F(x)= F(x)+x is odd function

∴-F(x)=F(-x), that is,-f (x)-x = f (-x)-x.

F(-x)=-f(x), then f(x) is odd function.

∴f(- 1)=-f( 1)=- 1

g(- 1)= f(- 1)+2 =- 1+2 = 1

If g(x) is a function with a period of 1 on R, then g(x)=g(x+ 1).

The function f(x)=x+g(x) happens to be a period in the interval [0, 1], and the range of interval length is [-2,5] ... (1).

Let x+ 1=t, and when x ∈ [0, 1], t = x+ 1 ∈ [1, 2].

Then f (t) = t+g (t) = (x+1)+g (x+1) = (x+1)+g (x)?

=[x+g(x)]+ 1？

Therefore, when t ∈ [1 2], f (t) ∈ [- 1 6] ... (2)

Similarly, let x+2=t, and when x ∈ [0, 1], t = x+2 ∈ [2 2,3].

Then f (t) = t+g (t) = (x+2)+g (x+2) = (x+2)+g (x)?

=[x+g(x)]+2

Therefore, when t ∈ [2 2,3], f (t) ∈ [0 0,7] ... (3)

According to the known conditions and (1)(2)(3), the range of f(x) in the interval [0,3] is [-2,7].

So the answer is: [-2,7].

From f(x+2)=f(x), we know that f(- 1/2)=f(3/2).

Therefore, f( 1/2)=f(3/2).

f(- 1/2)=f( 1/2)

So -a/2 + 1=(b/2 +2)/(3/2)

Finishing, 3a+2b=-2.

And f(- 1)=f( 1),

That is, 1-a=(b+2)/2,

Finishing, 2a+b=0.

So a=2 and b=-4.

a+3b=- 10

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