If 3f(x) + 2f(-x) = 5x − 10, what is the value of f(1)?

Question

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

GMATinsight · Accepted Answer

3f(x) + 2f(-x) = 5x − 10

@x=1, 3f(1) + 2f(-1) = 5 − 10 = -5

@x=-1, 3f(-1) + 2f(1) = -5 − 10 = -15

Multiply the equation 1 with 3 and equation 2 with 2 and subtract

5*f(1) = -15+30 = 15

f(1) = 3

Answer: Option D

pushpitkc · Answer

If x=1, 3f(1) + 2f(-1) = 5 − 10 = -5
If x=-1, 3f(-1) + 2f(1) = -5 − 10 = -15

Let f(1) = a and f(-1) = b
Now, the equations become :
3a + 2b = -5 
2a + 3b = -15 
Since we have been asked to find the value of f(1) or a, we have to eliminate f(-1) or b

Multiplying the first equation by 3 and subtracting from it the second equation(multiplied by 2)
3(3a + 2b) - 2(2a + 3b) = -15 + 30
9a - 4a = 15 -> 5a = 15 -> a(f(1)) =3 (Option D)

ScottTargetTestPrep · Answer

If we let x = 1, we have 3f(1) + 2f(-1) = 5(1) - 10, i.e., 3f(1) + 2f(-1) = -5.

If we let x = -1, we have 3f(-1) + 2f(-(-1)) = 5(-1) - 10, i.e., 3f(-1) + 2f(1) = -15.

Now we need to “get rid” of f(-1) to find the value of f(1). To do that, we multiply the first equation by 3 and the second equation by 2:

3 x   → 9f(1) + 6f(-1) = -15

2 x   → 6f(-1) + 4f(1) = -30

Now if we subtract 6f(-1) + 4f(1) = -30 from 9f(1) + 6f(-1) = -15, we have:

5f(1) = 15

f(1) = 3

Answer: D

rahul16singh28 · Answer

3f(x) + 2f(−x) = 5x − 10
2f(x) + 3f(-x) = -5x - 10

9f(x) + 6f(-x) = 15x - 30
4f(x) + 6f(-x) = -10x - 20

Subtract the above two equation -

5f(x) = 25x - 10
f(x) = 5x - 2
f(1) = 3

Hence, D.

pushpitkc · Answer

If  x=1 , the equation becomes  3f(1) + 2f(-1) = 5 - 10 = -5  -> (1)

If  x=-1 , the equation becomes  3f(-1) + 2f(1) = -5 - 10 = -15  -> (2)

3*(1) - 2*(2) ->  9f(1) - 4f(1) = -15 + 30 = 15  ->  5f(1) = 15

Therefore, the value of f(1) =  \frac{15}{5}  =  3(Option D)

CrackverbalGMAT · Answer

In functions questions like this one, the idea is to deal with the function as if you were dealing with variables in equations. So, we could treat f(x) and f(-x) as two variables, say a and b here. The given equation then becomes 3a + 2b = 5x -10.

However, this is still insufficient to find the value of f(1); to find f(1) we need the definition of f(x).

If we substitute -x in the place of x, f(x) becomes f(-x) and vice versa. Therefore, the given equation becomes, 3 f(-x) + 2 f(x) = -5x -10 (remember, we are substituting -x in the place of every x). This is nothing but 3b + 2a = -5x -10.

We now have two equations in terms of a, b and x viz,
3a + 2b = 5x – 10
2a + 3b = -5x -10.

Solving the equations above, we get, 5a = 25x – 10 or a = 5x – 2. This means f(x) = 5x – 2. Now that we know what f(x) is, finding f(1) is very easy.
f(1) = 5 x 1 – 2 = 3.

The correct answer option is D .

Hope that helps!

Ahmed9955 · Answer

for x=1 
3f(1) + 2f(-1) =5(1) -10 => -5 ........... (1)

For x=-1  
3f(-1) + 2f(- (-1)) =5(-1) -10
-> 2f(1)+ 3f(-1) = -15............(2)

consider  f(1) = m and f(-1) = n 
=>in (1) we have,
 3m + 2y = -5.......(A)
=> in (2) we have,
 2m + 3y = -15 ......(B)

Solve (A) and (B) for 'm'

we have m=3 => f(1) =3

IMO- D

Crytiocanalyst · Answer

For simplicity let us chose a and b 
3a + 2b = -5
2a + 3b = -15

3f(1) + 2f(-1) = 5 − 10 = -5
3f(-1) + 2f(1) = -5 − 10 = -15

Multiplying the 1 equation by 3 and 
Multiplying the 2nd equation by 2
2-1:

3(3a + 2b) - 2(2a + 3b) = -15 + 30
9a - 4a = 15 -> 5a = 15 
a=3

Therefore IMO D

Oppenheimer1945 · Answer

Looking at the functional equation it’s easy to prove that f in injective and linear.

Assume f(x)=ax+b
Solving 3f(x)+2f(-x)=5x-10
You get a=5, b=-2
f(x)=5x-2
f(1)=3

Posted from my mobile device

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If 3f(x) + 2f(-x) = 5x − 10, what is the value of f(1)?

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If 3f(x) + 2f(-x) = 5x − 10, what is the value of f(1)?

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