If f (x) and G (x) are odd and even functions on R respectively, and satisfy the x power of F (x) - G (x) = e, find the expression of F (x) g (x) .

If f (x) and G (x) are odd and even functions on R respectively, and satisfy the x power of F (x) - G (x) = e, find the expression of F (x) g (x) .

From F (x) - G (x) = e ^ x (1)
have to
f(-x)-g(-x)=e^(-x)
F (x) and G (x) are odd functions and even functions on R respectively
-f(x)-g(x)=e^(-x)（2）
(1) + (2) get
-2g(x)=e^x+e^(-x)
so
g(x)=-(e^x+e^(-x))/2
Substitute (1) to get
f(x)=(e^x-e^(-x))/2

If the odd function f (x) on R satisfies f (x + 2) = f (x), what is the value of F (6)

For odd functions, then f (0) = 0
f(6)
=f(4+2)
=-f(4)
=-f(2+2)
=f(2)
=f(0+2)
=-f(0)
=0

Let f (x) be an odd function defined on R. when x < 0, f (x) = x ^ 2 / 3, then f (8)=____

F (x) is an odd function defined on R, f (- x) = - f (x)
When x < 0, f (x) = x ^ 2 / 3,
Then f (8) = - f (- 8) = - (- 8) ^ 2 / 3 = - 64 / 3
Glad to answer for you
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If f (x) is an odd function defined on R and f (x + 4) = f (x), then f (8) = ()
A. 0B. 1C. 2D. 3

∵ f (x + 4) = f (x), ∵ f (8) = f (0), ∵ we know that f (x) is an odd function defined on R, ∵ f (0) = 0, that is, f (8) = f (0) = 0

Let f (x) be an odd function defined on R and f (4) + F (- 3) = 2, then f (3) - f (4)=______ ．

∵ f (x) is an odd function defined on R ∵ f (- x) = - f (x) ∵ f (4) + F (- 3) = f (4) - f (3) = 2, ∵ f (3) - f (4) = - 2, so the answer is: - 2

If f (x) is an odd function defined on R, f (X-2) = f (x) - f (2), f (1) = 1 / 2, then f (3) =?

f(1)=f(3)-f(2)=1/2 f(-1)=-f(1)=-1/2 f(-1)=f(1)-f(2)=-1/2 f(2)=1 f(3)=3/2

Let f (x) be an odd function defined on R, and if x > 0, f (x) = 2 × - 3, f (- 2) =

The solution of F (x) is an odd function
Then f (- x) = - f (x)
Then f (- 2) = - f (2) = - [2 × 2-3] = - 1

Given the odd function f (x) defined on R, if X & gt; 0 is f (x) = 3x-1, find the analytic expression of F (x)

x0
So f (- x) = 3 (- x) - 1 = - 3x-1
And because f (x) is an odd function
So f (x) = - f (- x) = 3x + 1
At the same time, f (x) is an odd function, so f (0) = 0
So f (x) is expressed as
f(x)=3x-1 x>0
3x+1 x

Given the odd function f (x) defined on (- ∞, + ∞), when x > 0, f (x) = 3x-1, find the analytic expression of F (x)

x0
So f (- x) = 3 (- x) - 1 = - 3x-1
For odd functions, then f (x) = - f (- x) = 3x + 1
The odd function has f (0) = 0
therefore
f(x)=
3x+1,x0

If f (x) is an odd function defined on R, when x > 0, f (x) = x2 + 3x, find the analytic expression of F (x)

x>0,f(x)=x^2+3x
x=0,f(x)=0;
x