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

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 [3f(1) + 2f(-1) = -5] → 9f(1) + 6f(-1) = -15

2 x [3f(-1) + 2f(1) = -15] → 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

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.

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)

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!

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

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

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