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Which of the following function follows the rule: f(a + b) = f(a)*f(b)

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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 10:30
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A. a^2+b^2+2ab+1 != a^2+b^2+2

B. 5^2(a+b)/3 != 5^2a/3 *5^2b/3 => 5^2(a+b)/9

C. 3(a+b)+2 != 3(a+b)+4

D. \(\sqrt{2(a+b)}\) != 2ab

E. 24^(a+b) = 24^a * 24^b => 24^(a+b)

So answer is E.
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 10:53
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)

A. f(x)=x^2+1
f(a+b) = (a+b)^2 + 1
f(a)*f(b) = (a^2 + 1)*(b^2 + 1)
clearly not equal. incorrect.


B. f(x)=(5^2x)/3
f(a+b) = (5^2(a+b))/3
f(a)*f(b) = (5^2a)/3 * (5^2b)/3 = (5^2(a+b))/9
Not equal. Incorrect.


C. f(x)=3x+2
f(a+b) = 3a + 3b +2
f(a)*f(b) = (3a+2)*(3b+2)
Not equal. Incorrect.


D. f(x)=sqrt(2x)
f(a+b) = sqrt(2(a+b))
f(a)*f(b) = sqrt(2a)*sqrt(2b) = 2*sqrt(ab)
Not equal. Incorrect


E. f(x)=24^x
f(a+b) = 24^(a+b)
f(a)*f(b) = (24^a)*(24^b) = 24^(a+b).
Equal. Correct.


Correct Answer: E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 10:55
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Correct Answer is E i.e. 24^x
Question is f(a+b) = f(a).f(b) example 2+2 = 2*2
Now plug in 4 in each function and plug in 2 in each function and square the result
Which ever option gives same answer is correct
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Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:04
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Let a = 2 and b=3

24^5=24^2 * 24^3

The answer is E

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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:05
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f(a+b)=f(a)∗f(b)

30 seconds approach
If you look at the answers and notice the Product Rule. it cut shorts the time
The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents.

2 min approach:Solving using number picking
lets assume a=2,b=2, a+b=4

f(a+b)=(4)^2+1 = 17
f(a)∗f(b)=(2^2+1)(2^2+1)=25 False

f(a+b)=[(5)^2*4]/3 = [5^8]/3
f(a)∗f(b)=[5^2*2]/3*[5^2*2]/3 False

f(a+b)=(4)+2 = 6
f(a)∗f(b)=(3(2)+2)(3(2)+2)=8*8 False

f(a+b)=√2*4=√8
f(a)∗f(b)=(√2*2)(√2*2)=√16 False

f(a+b)=\(24^4\)
f(a)∗f(b)=\((24^2)*(24^2)=24^4\) TRUE

Answer E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:05
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Definitely answer is E

The fight was between A and E

E is true in all.conditions

A fails at a=-1 and b=1

So hence answer is definitely E

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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:11
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It is obvious only option E will satisfy. Still the workings

A. f(x)=x^2+1 ; f(a+b) = (a+b)^2+1 and f(a)*f(b) = a^2*b^2 + a^2 + b^2 +1 - Not Equal

B. f(x)=5^2x/3 ; f(a+b) = 5^2(a+b)/3 and f(a)*f(b) = 5^2(a)/3*5^(b)/3 - Not Equal as denominator will be 9

C. f(x)=3x+2 ; f(a+b) = 3(a+b)+2 and f(a)*f(b) = 3(a+2)*3(b+2) - Not Equal

D. f(x)=√2x ; f(a+b) = √2(a+b) and f(a)*f(b) = 2√ab - Not Equal

E. f(x)=24^x ; f(a+b) = 24^(a+b) and f(a)*f(b) = 24^a*24^b=24^(a+b) - Equal

IMO E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:19
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

Method 1: As per the rule that needs to be true, we can foresee that if the variable is in exponentials then a mere multiplication can simply lead to addition and the rule can hold true for the equation.

As per this, we see that the options 2 and 5 are the only ones where the variable is completely in the exponentials.

In option 2 the denominator will not allow the rule to hold true as in the RHS it will multiply. Hence option 2 is not correct.

In option 5, we have no other issues and the variable in the exponentials will simply add up on multiplication of the entities and hence the rule will always hold true.

Now, we can plug in some sample values and see if our analysis is correct and mark the correct answer which is option E.

Method 2: Putting values. This method is fool proof but may take you a couple of minutes to solve this question.

Option 1: f(x)=x^2+1

f(1 + 1) = f(1) = 1^2 + 1 = 5
f(1) = 1^2 + 1 = 2

If rule is true here then,
f(1 + 1) = f(1) * f(1)

However,
5 <> 2

Hence,
f(1 + 1) <> f(1) * f(1)

Option 1 is not correct.

Option 2: f(x) = (5^(2x))/3

f(1 + 1) = f(2) = (5^(2*2)) / 3 = (5^4) / 3 = 625/3
f(1) = (5^(2*1)) / 3 = 25/3

If rule is true here then,
f(1 + 1) = f(1) * f(1)

625/3 = 25/3 * 25/3 -> Rule holds here.

f(1 + 0) = f(1) = (5^(2*1)) / 3 = 25/3
f(1) = (5^(2*1)) / 3 = 25/3
f(0) = (5^(2*0)) / 3 = 1/3

However,
25/3 <> 25/3 * 1/3

Now,
f(1 + 1) = f(1) * f(1);
BUT, f(1 + 0) <> f(1) * f(0);

Hence, rule holds true sometimes but not always for this option.

Option 2 is not correct.

Option 3: f(x) = 3x + 2

f(1 + 1) = f(2) = 3*2 + 2 = 8
f(1) = 3*1 + 2 = 5

If rule is true here then,
f(1 + 1) = f(1) * f(1)

However,
8 <> 5 * 5

Hence,
f(1 + 1) <> f(1) * f(1)

Option 3 is not correct.

Option 4: f(x) = sqrt(2*x)

Consider,
f(1 + 1) = f(2) = sqrt(2*2) = 2
f(1) = sqrt(2*1) = sqrt(2)

If rule is true here then,
f(1 + 1) = f(1) * f(1)

And,
2 = sqrt(2) * sqrt(2)

Now consider,
f(2 + 3) = f(5) = sqrt(2*5) = sqrt(10)
f(2) = sqrt(2*2) = 2
f(3) = sqrt(2*3) = sqrt(6)

However,
sqrt(10) <> 2 * sqrt(6)

Hence,
f(1 + 1) <> f(1) * f(1)

Option 4 is not correct.

Option 5: f(x) = 24^x

Consider,
f(1 + 1) = f(2) = 24^2
f(1) = 24^1

If rule is true here then,
f(1 + 1) = f(1) * f(1)

And,
24^2 = 24^1 * 24^1

Consider,
f(2 + 3) = f(5) = 24^5
f(2) = 24^2
f(3) = 24^3

If rule is true here then,
f(1 + 1) = f(1) * f(1)

And,
24^5 = 24^2 * 24^3

Hence Option 5 or Option E is correct.
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New post 08 Jul 2019, 11:20
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Answer : E

substituting a+b once for x in E yields
24^(a+b)
f(a)*f(b) = 24^aX24^b = 24^(a+b)

rest all equations yield different results. so E
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New post 08 Jul 2019, 11:22
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let a=1
b=2

E. f(a+b) = f(3)= 24^3
f (a) = 24
f (b) = 24^2
f (a) * f(b) = 24^3

IMO option E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 11:54
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f(a+b)=f(a)*f(b)
We know that x^(a+b)={x^a}*{x^b}. The function in the answer choices that most resemble this property is E. So I will test for this option first.
f(x)=24^x. f(a)=24^a and f(b)=24^b.
f(a+b)=24^(a+b)
f(a)*f(b)=(24^a)*(24^b) =24^(a+b).
Hence the answer is E.

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New post 08 Jul 2019, 12:16
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for \(f(x)=x^2+1\):
f(a+b) = \((a+b)^2 + 1 = a^2 + b^2 + 2ab + 1\)
f(a)∗f(b) = \(a^2 + 1 + b^2 + 1 = a^2 + b^2 + 2\) --> not equal for all values


for \(f(x)=\frac{5^{2x}}{3}\):
f(a+b) = \(\frac{5^{2(a+b)}}{3} = \frac{5^{2a}5^{2b}}{3}\)
f(a)∗f(b) = \(\frac{5^{2a}}{3}*\frac{5^{2b}}{3} = \frac{5^{2a}5^{2b}}{9}\)--> not equal for all values

for \(f(x)=3x+2\):
f(a+b) = \(3(a+b) + 2 = 3a + 3b + 2\)
f(a)∗f(b) = \((3a+2)(3b+2) = 9ab + 6b + 6a + 4\) --> not equal for all values

for f(x)= \(\sqrt{{2x}}\):
f(a+b) = \(\sqrt{{2(a+b)}} = \sqrt{{2a+2b}}\)
f(a)∗f(b) = \(\sqrt{{2a}}* \sqrt{{2b}} = \sqrt{{4ab}} = 2\sqrt{{ab}}\) --> not equal for all values

for \(f(x)= 24^x\):
f(a+b) = \(24^{a+b}\)
f(a)∗f(b) = \(24^{a}*24^{b} = 24^{a+b}\) --> equal

E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 12:20
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

A. f(x)=\(x^2+1\)

B. f(x)=\(5^2^x * 3\)

C. f(x)=3x+2

D. f(x)=\(\sqrt{2x}\)

E. f(x)=\(24^x\)

A - f(a) = \(a^2 +1\) f(b) = \(b^2 +1\) f(a+b) = \((a+b)^2 +1\).
f(a) * f(b) = \(a^2*b^2 +a^2+b^21\)
Clearly f(a+b) is not equal to f(a)∗f(b).

B - f(a)=\(5^2^a * 3\) f(b)=\(5^2^b * 3\) f(a+b)=\(5^2^(a+b) * 3\).
f(a) * f(b) = \(5^2^a+b * 9\)
Clearly f(a+b) is not equal to f(a)∗f(b).

C - f(a)=\(3a+2\) f(b)=\(3b+2\) f(a+b)=\(3(a+b)+2\).
f(a) * f(b) = \(9ab+6a+6b+4\)
Clearly f(a+b) is not equal to f(a)∗f(b).

D - f(a)=\(\sqrt{2a}\)
f(b)=\(\sqrt{2b}\)
f(a+b)=\(\sqrt{2(a+b)}\)
f(a)*f(b)=\(\sqrt{4ab}\)
Clearly f(a+b) is not equal to f(a)∗f(b).

E - f(x)=\(24^x\)

f(a)=\(24^a\)
f(b)=\(24^b\)
f(a+b)=\(24^(a+b)\)
f(a) * f(b)=\(24^(a+b)\).
Clearly f(a+b) is equal to f(a)∗f(b)

Hence E
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Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 13:28
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GMAT likes to throw such questions to waste your precious time.
It's not about concepts or clues, but about hard work on your pad.
Per my experience, the first two options are definitely the wrong answer :) you have to sweat searching for correct option :)

Let's begin

A) \((a^2+1)*(b^2+1)=(a+b)^2+1\); at a glance on the left side, we will have \(a^2*b^2\) that is impossible for the right hand. NO
B) \(5^{2a+2b}/9=5^{2a+2b}/3\); NO
C) \((3a+2)*(3b+2)=3(a+b)+2\); NO
D) \(\sqrt{4ab}=\sqrt{2a+2b}\) NO
Last one hold the breath, otherwise you have to go back checking all the options once again
E) \(24^a*24^b=24^{a+b}\)

IMO
Ans: E
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 13:53
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Quote:
Which of the following function follows the rule: f(a+b)=f(a)∗f(b)f(a+b)=f(a)∗f(b)?

A. f(x)=x2+1f(x)=x2+1

B. f(x)=52x3f(x)=52x3

C. f(x)=3x+2f(x)=3x+2

D. f(x)=2x‾‾√f(x)=2x

E. f(x)=24x


Here among all the options only in option e,

f(a+b) = f(a) * f(b)
as exponents add each other when multiplied.
base wont matter here.
Option E.
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 15:27
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

We have to look for the option in which addition and multiplication might result in same value.
If we look at the answer choices we can eliminate A, C and D easily. But to make sure we will put value of x and check.

A. f(x)=x^2+1
f(a+b)= (a+b)^2+1
f(a)*f(b) = (a^2 + 1)* (b^2+1)
after solving we do not get similar values.


B. f(x)=5^2x/3
f(a+b)=5^2(a+b)/3
f(a)*f(b)=5^2a/3* 5^2b/3
After solving this we get same numerator but different denominator.


C. f(x)=3x+2
f(a+b)=3(a+b)+2
f(a)*f(b)=(3a+2)*(3b+2)
after solving we do not get similar values.


D. f(x)=sq root of 2x
f(a+b)=sq root of 2(a+b)
f(a)*f(b)=sq root of 2a * sq root of 2b
after solving we do not get similar values.


E. f(x)=24^x
f(a+b)=24^(a+b)
f(a)*f(b)=24^a* 24^b
We get similar values on both sides.
Correct.

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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 16:31
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f(a+b) = f(a) • f(b)
So too many variables here let’s solidify it.
let a= -1 , b=1
.: f(-1+1) = f(-1)•f(1) —>
f(0)= f(-1)•f(1)

(a) f(x) = x^2+1
f(0)= 0^2+1 =1 and f(-1)•f(1)=
f(-1^2+1)•f(1^2+1) =4
So 1 is not equal to 2 (doesn’t follow rule)

(b) f(x)=((5)^2x)/3
f(0)=5^0/3 =1/3 and
f(-1)•f(1)=f(5^(-2)/3)•f(5^2/3)
= (1/3•25)•(25/3) =1/9
1/3 not equal to 1/9

(c) f(x) = 3x+2
f(0)=3(0)+2 =2 and f(-1)•f(1)= f(3(-1)+2)•f(3(1)+2)= -5
Doesn’t follow rule

(d) f(x) = (sqrt.2x)
f(0)=( sqrt.2(0))= 0 and
f(-1)•f(1)= f(sqrt.2(-1))•f(sqrt2(1)) =2i
Doesn’t follow rule

(e) f(x) = 24^x
f(0) = 24^0 = 1 and f(-1)•f(1)= f(24^-1)•f(24)= 1
1 equals 1 follows rule of f(0) = f(-1)•f(1)

Answer E

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Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 16:37
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\(24^x\) where a = 2, b= 2.

\(f(a + b) = f(a) * f(b)\)

\(f(2 + 2) = f(2) * f(2)\)

\(4 = 4\)

For \(f(a + b)\)

\(24^4 = (3^4 * 2^12)\)

For \(f(a) * f(b)\)

\(24^2 * 24^2\)

\((2^6 * 3^2) * (2^6 * 3^2)\)

\((2^12 * 3^4)\)

E
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New post 08 Jul 2019, 18:47
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A. f(x)=x^2+1, if we replace, we have (a+b)^2 +1 ≠ (a^2+1)*(b^2 +1)

B. f(x)=5^2x/3, if we replace, we have 5^2(a+b)/3 ≠ (5^2a/3)*(5^2b/3)

C. f(x)=3x+2, if we replace, we have 3(a+b)+2 ≠ (3(a+2))*(3(b+2))

D. f(x)=√2x, if we replace, we have √2(a+b) ≠ √2a*√2b

E. f(x)=24x, if we replace, we have 24^(a+b) = 24^a*24^b

So (E) is our answer.
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)  [#permalink]

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New post 08 Jul 2019, 20:01
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IMO - E

Because 24^x

for f(a+b) = 24^(a+b) = 24^a*24^b
for f(a)*f(b) = 24^a*24^b.
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)   [#permalink] 08 Jul 2019, 20:01

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