Determine f(x) if 2*f(x) + 3*f(1-x) = 4x - 1

We conclude that the function we have to determine is a
linear function, because the result of the sum of the functions is a linear function 4x
- 1.

We'll substitute x by 1-x in the given
relation.

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

We'll remove the brackets form the right
side:

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

We'll combine like
terms:

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

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

We'll eliminate the unknown f(1-x). For this reason,
we'll multiply (1) by 3 and (2) by -2:

6f(1-x) + 9f(x) = 9
- 12x (3)

-6f(1-x) - 4f(x) = -8x + 2
(4)

We'll add (3) +
(4):

6f(1-x) + 9f(x) - 6f(1-x) - 4f(x) = 9 - 12x - 8x +
2

We'll eliminate and combine like
terms:

5f(x) = -20x + 11

We'll
divide by 5:

The function
is:

f(x) = -4x +
11/5

Another method of solving the problem
is to consider the linear function:

f(x) = ax +
b

f(1-x) = a(1-x) + b

f(1-x) =
a - ax + b

We'll re-write the expression 2*f(x) + 3*f(1-x)
= 4x - 1

2ax + 2b - 3ax + 3a + 3b = 4x -
1

We'll combine like
terms:

-ax + 3a + 5b = 4x -
1

The expressions from both sides are equal if the
correspondent coefficients are equal.

-a =
4

a =
-4

3a + 5b =
-1

3*(-4) + 5b = -1

-12 + 5b =
-1

5b = 11

b =
11/5

f(x) = -4x +
11/5