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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

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

for x = 1

=> f(3) + 3*f(-3) = 1 + 1 = 2 ...(1)

for x = -1

=> f(-3) + 3*f(3) = 0 ...(2)

(1) - 3*(2)

=> f(3) + 3*f(-3) - 3*f(-3) - 9*f(3) = 2

=> f(3) - 9*f(3) = 2

=> -8*f(3) = 2

=> f(3) = -1/4

The required value of f(3) = -1/4

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll create a new relation, substituting x by -x.

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

We'll put f(3x)+3*f(-3x)=x+1 (1)

We'll determine f(3x) from the system formed by the relations (1) and (2):

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

We'll substitute (3) in (1):

f(3x) + 3[ -x + 1 - 3*f(3x)] = x + 1

We'll remove the brackets:

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

We'll combine like terms:

-8f(3x) = 4x - 2

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

We'll substitute x by 1 and we'll get f(3):

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

f(3) = -1/4

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