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Re: Which of the following function follows the rule: f(a + b) = f(a)*f(b)
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08 Jul 2019, 10:19
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.