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