If f(a*b) = f(a) + f(b), what is the value of f(1)?

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Sun Jan 06, 2019 11:43 pm

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[Math Revolution GMAT math practice question]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

A. 1
B. -1
C. 0
D. 2
E. -2

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Source: Beat The GMAT — Problem Solving | Sun Jan 06, 2019 11:43 pm

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Jan 07, 2019 3:16 am

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

A. 1
B. -1
C. 0
D. 2
E. -2

$$f\left( {ab} \right) = f\left( a \right) + f\left( b \right)\,\,\,\,\,\left( * \right)$$
$$? = f\left( 1 \right)$$
$$f\left( 1 \right) = f\left( {1 \cdot 1} \right)\mathop = \limits^{\left( * \right)} f\left( 1 \right) + f\left( 1 \right)$$
$$f\left( 1 \right) = 2 \cdot f\left( 1 \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = f\left( 1 \right) = 0$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 07, 2019 3:42 am

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

A. 1
B. -1
C. 0
D. 2
E. -2

Let a=1 and b=2.
Plugging a=1 and b=2 into f(ab) = f(a) + f(b), we get:
f(1*2) = f(1) + f(2)
f(2) = f(1) + f(2)
0 = f(1).

The correct answer is C.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Jan 07, 2019 6:52 am

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

A. 1
B. -1
C. 0
D. 2
E. -2

To generalize Mitch's and Fabio's solutions, notice that, as long as we can get f(1) involved in the process, we'll arrive at answer choice C every time.

Notice that k = 1*k
Since f(a*b) = f(a) + f(b), we can write: f(1*k) = f(1) + f(k)
Simplify: f(k) = f(1) + f(k)
Subtract f(k) from both sides to get: 0 = f(1)

Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Jan 09, 2019 12:04 am

=>

Since 1 = 1 * 1, setting a = 1 and b = 1 yields f(1) = f(1*1) = f(1) + f(1) and f(1) = 2f(1), which is equivalent to f(1) = 0.

Therefore, the answer is C.
Answer: C

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8134; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Jan 21, 2019 6:00 pm

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

A. 1
B. -1
C. 0
D. 2
E. -2

We can let both a and b be 1, and thus we have:

f(1*1) = f(1) + f(1)

f(1) = 2f(1)

Subtracting f(1) from both sides, we have:

0 = f(1)

Alternate Solution:

We can let a be 1, and thus we have:

f(1*b) = f(1) + f(b)

f(b) = f(1) + f(b)

Subtracting f(b) from both sides, we have:

0 = f(1)

Answer: C

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