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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, 20:19
1
We need to check every option whether they satisfy.

A. \(f(x) = x^2 + 1\)
So,
\(f(a + b) = (a + b)^2 + 1 = a^2 + 2ab + b^2 + 1\)
\(f(a) * f(b) = (a^2 + 1) * (b^2 + 1) = a^2 + b^2 + a^2b^2 + 1\)
Clearly f(a + b) Not Equal to f(a)*f(b)

Similarly, it will not satisfy for options B, C, D

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

\(f(a + b) = 24^{a + b} = 24^a*24^b\)
\(f(a) * f(b) = 24^a*24^b\)
So, f(a + b) = f(a) * f(b)
Satisfies.

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, 20:25
1
this is substitution kind of question. Just substitute each answer choice with a+b and find if we get the output of 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 both are not equal.



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

similarly, f(a+b)=5(a+b)^2/3, f(a).f(b) = 5^2a/3 . 5^2b/3
both are not equal.



C. f(x)=3x+2

f(a+b) = 3(a+b) + 2 , f(a). f(b) = (3a+2).(3b+2). these exp are also not same.

D. f(x)=√2x

f(a+b) = √2(a+b). f(a).f(b) = √2(a).√2(b), which are not equal.

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

Correct choice 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, 20:30
1
take a =1, b=1

A. f(x)=x^2+1 gives 5 and 4 hence wrong

B. f(x)=5^2 * x/3 gives different ans hence wrong

C. f(x)=3x+2 gives 8 and 25 hence wrong

D. f(x)=\sqrt{2x} gives 2 and 2 hence keep

E. f(x)=24^x gives 24^2 and 24^2 hence keep

now substitute 1 and 4 for d and e

d) gives \sqrt{10} and 4 hence wrong

e) gives 24 ^5 and 24 ^5. hence e is 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, 20:48
1
f(a+b) = f(a) * f(b)

A. f(x)=x^2+1
replace LHS f(x) = f(a+b) RHS - (a+b)^2 + 1
RHS f(a) = a^2 & f(b) = b^2 -> RHS is not equal to LHS

B. f(x)=5^2 / 3
LHS f(a+b) = 5 (a+b)^2 / 3
RHS f(a) = 5a^2/3 f(b) = 5b^2/3
not equal

C. f(x)=3x+2
f(a+b) = 3(a+b)+2
f(a) * f(b) = (3a+2) (3b+2)

D. f(x)=√2x‾‾
f(a+b) = √2(a+b)‾‾
f(a) = √2a‾‾ + √2b‾‾ = √2‾‾ (√a‾‾ + √b‾‾ )
not equal

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

finally a match , 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, 21:00
1
F(a+b)= f(a)*f(b)
F(a+b)= 24^x * 24^x = 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 08 Jul 2019, 21:02
1
My answer E
Only E if we subsitute yeilds equality between both sides. We can plug in numbers to see if this is true. Pick easy numbers to work with, such that a=5, b=5 OR a=2, b=8.
E. 24^10=24^5*24^5=24^5+5=24^10.
OR 24^10=24^2*24^8=24^10.
Wer can go ahead and try substituting values for other questions, but none will work, I checked.
Thus 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, 21:37
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by option verification method we can substitute the values i.e f(x)=24power x
24 power(a+b)=24 power a*24 power b
since bases are powers must be added it becomes 24 power(a+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, 21:57
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by putting a values a=2 , b=1 we can eliminate option C,D.

Thn put value a=3,b=2 we can eliminate option A,B.

Hence 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, 22:02
2
Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

A. \(f(x) = x^2+1\)
\(f(a + b) = (a + b)^2 + 1 = a^2 + b^2 + 2ab + 1\)
\(f(a)*f(b) = (a^2+1)*(b^2+1) = (ab)^2 + a^2 + b^2 + 1\) --> NO

B. \(f(x) = \frac{5^{2x}}{3}\)
\(f(a + b) = \frac{5^{2(a + b)}}{3} = \frac{5^{2a + 2b}}{3}\)
\(f(a)*f(b) = \frac{5^{2a}}{3}*\frac{5^{2b}}{3} = \frac{5^{2a + 2b}}{9}\) --> NO

C. \(f(x) = 3x + 2\)
\(f(a + b) = 3(a + b) + 2 = 3a + 3b + 2\)
\(f(a)*f(b) = (3a + 2)*(3b + 2) = 9ab + 6a + 6b + 4\) --> NO

D. \(f(x) = 2x\)
\(f(a + b) = 2(a + b) = 2a + 2b\)
\(f(a)*f(b) = 2a*2b = 4ab\) --> NO

E. \(f(x) = 24^x\)
\(f(a + b) = 24^{a + b}\)
\(f(a)*f(b) = 24^a*24^b = 24^{a + b}\) --> YES

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, 22:14
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Lets check answer choices.

(A)f(x)=\(x^{2}+1\)
f(x+1)=\((x^{2}+2x+1)+1\)
f(1)=2
Clearly \(f(x+1)\neq f(x)*f(1)\)

(B)f(x)=\(\frac{5^{2x}}{3}\)
f(x+1)=\(\frac{5^{2(x+1)}}{3}\) =\(\frac{5^{2(x)}}{3}\) *5
f(1)=\(\frac{5^{2(1)}}{3}\)
Clearly \(f(x+1)\neq f(x)*f(1)\)
But this choice tells us that, had there been no coefficient (1/3) this choice would have been correct. So we are looking for choice, which has x in the power, and there are no co-efficient/constants multiplied to the base.
Since, \(a^{x+y}\) =\(a^{x}*a^{y}\)

Glancing through other options, 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, 23:06
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Here, we can just put different values for a and b verify. (simple way)

(A) Here, +1 in the function will create issue when doing f(a)*f(b).

(B) 3 in the denominator will create issue when doing f(a)*f(b).

(C) + 2 in the function will create issue here.

(D) square root of x and 2x will differ by factor of 1.4 (root 2). (try putting a and (a+a). So, f(a)*f(a) is not equal to f(2a)).

(E) this function follows the given property.


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, 23:34
Which of the following function follows the rule: f(a+b)=f(a)∗f(b)f(a+b)=f(a)∗f(b)?

Correct answer is A

This one is easy because when you are looking for multiplication of two quantities to be equal to the addition of those two quantities, think about 1.
In this case, the very first answer choice has something to do with 1, so if we evaluate that, we can see that it satisfies the rule in the question.

x^2 + 1 = x^2 * 1
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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, 00:13
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Suppose a = 2 and b = 3
Then f(a+b) = f(5)

Only option E is correct and satisfies the Equality f(a+b) = f(a) * f(b)
when we put the values of a and b
f(a)= f(2) = 24^2
f(b) = f(3) = 24^3
f(a+b) = f(5) = 24^5
f(a)*F(b) = f(2) *f(3) = (24^2)*(24^3)= 24^5
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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, 00:17
Which of the following function follows the rule: f(a+b)=f(a)∗f(b)f(a+b)=f(a)∗f(b)?

this question can eat up your time a lot if you don't find a pattern
A. f(x)=x^2+1 : plug in
F(x)= f(A+B)= (A+B)^2
f(A)*f(B)= (A)^2 *B^2 : not equal no notice that we have AxB on one side and A+B on other to equalize them we need A and B in power : hints of A^n x A^r= A(n+r)

B. f(x)=(5^2x)/3 hold


C. f(x)=3x+2: eliminate x not in power

D. f(x)=(2x)^.5: eliminate :x not in power

E. f(x)=24^x:hold


plug A and B again
E first as its easier : f(A+B) :24^A+B , f(A)*f(B) = 24^(A) *24^(B) both are equal so E

no need to check 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 09 Jul 2019, 00:39
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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, 00:51
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Option E is the right answer. f(a+b)=24^(a+b)=(24^a)*(24^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 09 Jul 2019, 01:01
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Here we can apply POE:

A. f(a)=a^2+1, f(b)=b^2+1,
f(a+b) =(a+b)^2+1 = (a^2+1)*(b^2+1) => WRONG

B. f(a)=5^2a/3, f(b)=5^2b/3
f(a+b)=5^2(a+b)/3=5^2(a+b)/3*3 => WRONG

C. f(a)=3a+2, f(b)=3b+2
f(a+b) = 3(a+b)+2 = (3a+2)*(3b+2)

D. f(a)=√2a, f(b)=√2b
f(a+b)=√2(a+b)=√4ab => WRONG

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

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 09 Jul 2019, 01:41
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A) f(a+b)=(a+b)^2+1=(a^2+1)*(b^2+1)
a^2+2ab+b^2+1=a^2b^2+a^2+b^2+1
2ab does NOT equal to a^2b^2 (NO)
B) (5^2a+2b)/3=(5^2a)/3*(5^2b)/3
(5^2a+2b)/3 does NOT equal (25^2a+2b)/9. OUT
C) 3(a+b)+2=(3a+2)(3b+2)
3a+3b+2=9ab+6a+6b+4
0 does NOT equal 9ab+3a+3b+2. OUT
D)sqr root 2(a+b)=sqr root 2a*sqr root 2b
sqrt 2a+2b does NOT equal sqrt 4ab. OUT
E) 24^a+b=24^a*24^b
24^a+b=24^a+b. YES, this is 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 09 Jul 2019, 02:50
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when we substitute the value in E
\(24^a*24^b\)=f(a+b)
in no other case this function is same. So 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 09 Jul 2019, 03:19
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Which of the following function follows the rule: f(a+b)=f(a)∗f(b)?

We can bruteforce this question by input a sample or imagine a smarter way.
What operation a+b is given to us by a*b. It reminds me about power.
x^(a+b) = (x^a)*(x^b)
So let's check first of all powers, that is variants B and E.
E seems the easiest one, so let's start from E.
Sample a = 2 , b =3
E. 24^(5) = (24^3)*(24^2) True

Amsw is 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, 03:19

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