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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, 07:53
1
IMO E


1: let x=1,b=2
f(a+b)=(1+2)^2+1=10,
f(a)+f(b)=(1+1)+( 4+1)=7 so this option is wrong

2:let a=0 and b=1
(5^2(1))/3 not eqaul to (5^2(0))/3 + (5^2(1))/3
wrong

3: let a=0 and b=1
(3(0)+2)+(3(1)+2)=7
and 3(1)+2=5
and so this option is also wrong

4:let a=1 and b=2
root(2(1))+root(2(2))=2+root(2)
root(2)(3)=root(6)

so f(a+b) not equal to f(a)+(b) and this option is wrong for this reason.

5:let a=1 and b=2
(24)^(3)=24^1*24^2

Option E is the correct 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, 08:08
1
IMO E.

f(a+b)=f(a)∗f(b).
Let's consider a=2 and b=3 for all the options.

1. f(x)=x^2+1
f(2)= 5 and f(3) = 10. f(2+3)= 26 =/= 50. Incorrect option.

2. f(x)=5^(2x)/3
f(2)= 5^(4)/3 and f(3)= 5^(6x)/3. f(5) = 5^(10x)/3 =/= 5^(10x)/9. Incorrect option.

3. f(x)=3x+2
f(2)= 8, f(3)= 11, f(5)=17 =/= 88. Incorrect option.

4. f(x)=√2x
f(2)= 2, f(3)= √6, f(5)= √10 =/= 2√6. Incorrect option.

5. f(x)=24^x.
f(2) =24^2 , f(3)=24^3, f(5) = 24^5 ==24^5. Correct option. .
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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 Updated on: 08 Jul 2019, 08:21
1
Which of the following function follows the rule: f(a+b)=f(a)∗f(b)



\(A. f(x)=x^2+1\)
\(f(a+b)= (a+b)^2+1\)
\(f(a)= a^2+1\)
\(f(b)= b^2+1\)
\(f(a).f(b)= (a^2+1)(b^2+1)\)
f(a+b)<>f(a).f(b)

B. f(x)=5^2x/3
f(a+b)= 5^2(a+b)/3
f(a)= 5^2a/3
f(b)=5^2b/3
f(a).f(b)= 5^2(a+b)/9
f(a+b)<>f(a).f(b)


C. f(x)=3x+2
f(a+b)= 3(a+b)+2 = 3a+3b+2
f(a)= 3a+2
f(b)= 3b+2
f(a).f(b)= (3a+2)(3b+2) = 9ab+6b+6a+4
f(a+b)<>f(a).f(b)


\(D. f(x)=\sqrt{2x}\)
\(f(a+b)= \sqrt{2(a+b)}\)
\(f(a)=\sqrt{2a}\)
\(f(b)=\sqrt{2b}\)
\(f(a).f(b)=2\sqrt{ab}\)
f(a+b)<>f(a).f(b)


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


IMO E
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Originally posted by Kinshook on 08 Jul 2019, 08:15.
Last edited by Kinshook on 08 Jul 2019, 08:21, edited 2 times in total.
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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, 09:53
1
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, 10:19
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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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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, 11:16
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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, 11:20
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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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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, 14:27
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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, 23:39
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

Let us try a=1, b=3, a+b= 1+3 = 4

A. f(x)=x^2+1
- f(a+b)=f(4)=17
- f(a)=f(1)=2, f(b)=f(3)=10, then f(a)∗f(b)= f(1)∗f(3) = 20
- the above function does NOT follow the rule: f(a+b)=f(a)∗f(b)

B. f(x)=5^(2x)/3
- f(a+b)=f(4)=(5^8)/3
- f(a)=f(1)=(5^2)/3, f(b)=f(3)=(5^6)/3, then f(a)∗f(b)= f(1)∗f(3) = (5^8)/9
- The above function does NOT follow the rule: f(a+b)=f(a)∗f(b)

C. f(x)=3x+2
- f(a+b)=f(4)=14
- f(a)=f(1)=5, f(b)=f(3)=11, then f(a)∗f(b)= f(1)∗f(3) = 55
- the above function does NOT follow the rule: f(a+b)=f(a)∗f(b)

D. f(x)=√(2x)
- f(a+b)=f(4)=√8
- f(a)=f(1)=√2, f(b)=f(3)=√6, then f(a)∗f(b)= f(1)∗f(3) = √12
- the above function does NOT follow the rule: f(a+b)=f(a)∗f(b)

E. f(x)=24^x
- f(a+b)=f(4)=24^4
- f(a)=f(1)=24^1, f(b)=f(3)=24^3, then f(a)∗f(b)= f(1)∗f(3) = 24^4
- the above function does follow the rule: f(a+b)=f(a)∗f(b)

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 09 Jul 2019, 03:48
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A. f(x)=\(x^2\)+1

f(a+b) = \(a^2\)+\(b^2\)+2ab+1
f(a).f(b) = \(a^2\)\(b^2\)+\(a^2\)+\(b^2\)+1

Not equal

B. f(x)=\(5^{2x}\)/3

f(a+b) = \(5^{2a+2b}\)/3
f(a).f(b) = \(5^{2a+2b}\)/9

Not equal

C. f(x)=3x+2

f(a+b) = 3a+3b+2
f(a).f(b) = (3a+2)(3b+2)

Not equal

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

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

Not equal

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

f(a+b) = \(24^{a+b}\)
f(a).f(b) = \(24^{a+b}\)

Equal

Hence, option E.
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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)   [#permalink] 09 Jul 2019, 03:48

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