Find 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.

f(x)=ax+b.

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

You need to consider the function `f(x) = mx + n` , hence `f(-x) = -mx + n` .

Using the information provided by the problem yields:

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

`mx + n - 2(-mx + n) = 3x - 1`

`mx + n + 2mx - 2n = 3x - 1`

`3mx - n = 3x - 1`

Equating the coefficients of like powers yields:

`{(3m = 3),(-n = -1):} => {(m = 1),(n = 1):}`

**Hence, evaluating the function f(x), under the given conditions, yields **`f(x) = x + 1.`

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

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