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

We notice that f(x) is a linear function:

==> let f(x) = ax+b

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

Now substiute :

==> 2(ax+b) + 3(-ax+a+b) = 4x - 1

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

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

**==> a= 4**

==> 3a + 5b = -1

==> 12 + 5b = -1

==> 5b = 11

**==> b= 11/5**

**==> f(x) = 4x + 11/5**

Here we have to find f(x). We are given that 2*f(x) + 3*f( 1-x) = 4x-1

Let's take f(x) as ax + b

So f(x) = ax+b

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

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

=> 2*(ax+b) + 3*[a(1-x) + b] = 4x -1

=> 2ax + 2b + 3[ a- ax + b] = 4x - 1

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

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

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

=> -a = 4 and 3a + 5b =-1

Now substituting a= -4 in 3a + 5b =-1

=> -12 + 5b =-1

=> 5b = 11

=> b =11/5

**So the function f(x) = -4x + 11/5 **

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**

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

Solution:

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

Replace x by 1-x :

2f(1-x) + 3f(1-[1-x]} = 4(1-x) -1 .

Simply:

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

Eq(1)*2-eq(2)*3 eliminates f(1-x) and we solve for f(x)"

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

4f(x) - 9f(x) = 8x-2-9 + 12x = 16x-11

-5f(x) = 20x-11

f(x) = (20x-11)/-5 = (11-20x)/5 = 2.2-4x

f(x) = 2.2-4x