# What is f(x) if f(x)-2f(-x)=3x-1

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As we can notice, the given expression f(x)-2f(-x)=3x-1 is a linear function.

We'll substitute "x" by "-x" and we'll rewrite the above expression.

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

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

-2f(-x)+f(x)=3x-1 (2)

We'll consider f(x) and f(-x) as unknowns.

We'll multiply the expression (1) by the value "+2" and after that we'll add the expression (1) to the expression (2).

2f(-x)-4f(x)-2f(-x)+f(x)=-6x-2+3x-1

-3f(x)=-3x-3

We'll divide the expression by "-3"

**f(x)=x+1**

where a=1 and b=1

To find f(x) if f(x) - 2f(-x) = 3x-1.

Solution:

f(x) -2f(-x) = 3x-1....(1)

Replace x by -x:

f(-x) -2f(-(-x) ) = 3(-x) -1.

f(-x) -2f(x) = -3x-1...(2)

Eq(1)*2+ Eq(2) gives:

2{f(x)-2(-x)}+f(-x) -2f(x) = 2(3x-1) - 3x-1

-4f(-x) +f(-x) = 6x-2 -3x-1 = 3x-3

-3f(-x) = 3x-3

f(-x) = 1-x.

Therefore f(x) = 1-(-x) = x+1.

Therefore f(x) = x+1

We have to find f(x) given that f(x) - 2f(-x) = 3x - 1

Now if we replace x with -x we see that

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

Adding f(x) - 2f(-x) = 3x - 1 and f(-x) - 2f(x) = -3x -1

=> f(x) - 2f(-x) + f(-x) - 2f(x) = 3x - 1 -3x -1

=> -f(x) - f(- x) = -2

=> f(x) + f(-x) = 2

f(-x) = 2- f(x)

substitute this in f(x) - 2f(-x) = 3x - 1

=> f(x) - 2*[ 2- f(x)] = 3x -1

=> f(x) - 4 + 2f(x) = 3x -1

=> 3f(x) = 3x + 3

**Therefore f(x) = x +1**