If f(x) is a linear function, what is the value of f(1)?

Question

(1) f(x − 3) = f(x) + 1
(2) f(5) = 2f(3)

pushpitkc · Accepted Answer

If f(x) is a linear function, it is equal to ax+b.

1. f(x-3) = f(x) + 1
 a(x-3) + b = ax + b + 1
ax - 3a + b = ax + b + 1
-3a = 1
a = -1/3, but we can't determine value of b(Insufficient)

2. f(5) = 2f(3)
 5a +b = 2(3a + b)
5a + b = 6a + 2b
-a = b
Hence f(1) = a(1) + b = a - a = 0(Sufficient)  (Option B)

IanStewart · Answer

The algebraic solution pushpitkc provides above is good, and I imagine that's how most people will solve this question. There is a non-algebraic solution too, but it requires a very good understanding of slopes in coordinate geometry:

Rewriting Statement 1, we know that f(x) = f(x-3) - 1. That just means when we move three units to the right, from x-3 to x, the line falls by 1 unit. So the slope of the line is -1/3. We still have no idea how high or low the line is in the coordinate plane, so we can't locate any points on the line.

From Statement 2, we know that f(5) = 2f(3), so we know that f(5) = f(3) + f(3). So when the line moves two units to the right from f(3), it goes up by f(3). Since we have a line, which has a constant slope, then if we move two units to the  left  of f(3), to f(1), the line must therefore fall by f(3). But then the line will be at f(3) - f(3) = 0. So f(1) = 0 must be true, and Statement 2 is sufficient alone.

LeoN88 · Answer

IMG_20190602_213114__01.jpg

CrackverbalGMAT · Answer

An excellent question on functions, although there is a tiny bit of co-ordinate geometry and algebra involved.

The general form of a linear expression is ax + b. This is exactly what the question statement is trying to give as information. If you miss this bit, you may end up thinking that both statements are required together to solve the question.

So, let f(x) = ax + b where a and b are constants and a≠0. If we need the value of f(1), we will have to find out the values of a and b.

From statement I alone, we can say, 
a(x-3) + b = ax + b + 1, which on simplification, yields, 
ax – 3a = ax + 1.
Simplifying the above equation, we get a = - \frac{1}{3} . But, we do not know the value of b.

In co-ordinate geometry terms, this means that we know the slope of the line but we do not know its y-intercept. This essentially means that we do not know the equation of the line. 
 f(x) always represents a unique point on a line/curve.  
Because we do not know the equation of the line, it’s not possible to find a unique value for f(1).

Statement I alone is insufficient. Options A and D can be eliminated. Possible answers are B, C and E.
 
From statement II alone, we know f(5) = 2 f(3). This means, 
5a + b = 2(3a + b), which simplifies to a = -b. Plugging in the value of a in the original equation for the function, we have, f(x) = -bx +b which yields f(x) = b(x-1). 
From this, we can definitely say f(1) = b(1-1) = 0. Therefore, statement II alone is sufficient. The correct answer option is B.

Remember, in a DS question, it's important to test out the individual statements before combining. If you want to do this more often rather than getting into the trap of subsconsciously combining the statements, you have to learn to analyse the question statements and question data more.

Hope this helps!

ScottTargetTestPrep · Answer

Question Stem Analysis:

We are told that f is a linear function, and we need to determine the value of f(1). Since f is a linear function, f must be of the form f(x) = mx + b.

Statement One Alone:

\Rightarrow  f(x − 3) = f(x) + 1

Since f(x) = mx + b, f(x - 3) = m(x - 3) + b = mx - 3m + b. Then:

\Rightarrow  f(x - 3) = f(x) + 1

\Rightarrow  mx - 3m + b = mx + b + 1

\Rightarrow  -3m = 1

\Rightarrow  m = -1/3

Using this statement, we can determine that m = -1/3, however, without the value of b, we cannot answer the question.

If f(x) = -x/3, then f(x - 3) = -(x - 3)/3 = (3 - x)/3, and f(x) + 1 = -x/3 + 1 = -x/3 + 3/3 = (3 - x)/3. We see that f(x - 3) = f(x) + 1 is satisfied. In this case, f(1) = -1/3.

On the other hand, if f(x) = -x/3 + 1, then f(x - 3) = -(x - 3)/3 + 1 = -x/3 + 2, and f(x) + 1 = -x/3 + 1 + 1 = -x/3 + 2. Once again, f(x - 3) = f(x) + 1 is satisfied. However, in this case, we have f(1) = -1/3 + 1 = 2/3.

Statement one alone is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

\Rightarrow  f(5) = 2f(3)

We have f(5) = 5m + b, and 2f(3) = 2(3m + b) = 6m + 2b. Since we are told that these two quantities are equal, we can write:

\Rightarrow  f(5) = 2f(3)

\Rightarrow  5m + b = 6m + 2b

\Rightarrow  -b = m

Thus, f(x) = mx + b = -bx + b. It follows that f(1) = -b + b = 0. Statement two alone is sufficient.

Answer: B

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If f(x) is a linear function, what is the value of f(1)?

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