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

f(a+b)=f(a)∗f(b)

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

f(a+b) = 24^(a+b)

f(a)∗f(b)= \(24^a\)*\(24^b\)=24^(a+b)....(properties of exponent)

hence in option E f(a+b)=f(a)∗f(b) answer

f(a+b)=f(a)∗f(b)

Lets assume values for a and b.

Let a = 2 and b =3

A. f(x)=x^2+1
f(2) = 5, f(3) = 10
f(2)*f(3) = 50
f(2+3) = f(5) = 26
f(2)*f(3) not equals f(2+3)
Therefore not right

B. f(x)=5^2x/3
f(2) = 5^4/3, f(3) = 5^6/3
f(2)*f(3) = (5^4 * 5^6)/(3*3) = 5^10/9
f(5) = 5^10/3
f(2)*f(3) not equals f(2+3)
Therefore not right

C. f(x)=3x+2
f(2) = 8, f(3) = 11
f(2)*f(3) = 88
f(5) = 17
f(2)*f(3) not equals f(2+3)
Therefore not right

D. f(x)=√2x
f(2) = \(\sqrt{4}, f(3) = \sqrt{6}\)
f(2)*f(3) = \(\sqrt{24}\)
f(5) = \(\sqrt{10}\)
f(2)*f(3) not equals f(2+3)
Therefore not right


E. f(x)=24^x
f(2) = 24^2, f(3) = 24^3
f(2)*f(3)= 24^5
f(5) = 24^5

f(2)*f(3) equals f(2+3)

Therefore E is the answer.

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

A. f(x)=x2+1

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

C. f(x)=3x+2

D. f(x)=√(2x)

E. f(x)=24^x --> correct: f(a+b) = 24^(a+b) = 24^a * 24^b = f(a)*f(b)

f(x)=x^2+1 - not correct f(a+b) = (a+b)^2 + 1 which is not equal to (a^2+1)(b^2+1)

B. f(x)=(5^(2x))/3 - not correct f(a+b) = (5^(2a+2b))/3 not equal to (5^(2a))/3 * (5^(2b))/3

C. f(x)=3x+2 - not correct f(a+b) = 3(a+b) + 2 not equal to (3a+2)*(3b+2)

D. f(x)=sqrt(2x) not correct sqrt (a+b) is not equal to sqrt(a)*sqrt(b)

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

IMO E;
for the given function of the set ; f(a+b)=f(a)∗f(b)f(a+b)=f(a)∗f(b)
substitute value of function of x for option E ; f(x)=24x
we get 24^(a+b) = 24^a*24^b


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

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
---> 24^(a+b)=24^a*24^b=24^(a+b)
Answer choice is E.

Simple substitution gives:
A. f(a+b) = (a+b)^2+1
f(a)*f(b) = (a^2+1)*(b^2+1)
Since both are not equal, A is Wrong.

B. f(a+b) = (5^2(a+b))/3
f(a)*f(b) = (5^2(a+b))/3
Since both are equal, B is Correct.

C. f(a+b) = 3(a+b) +2
f(a)*f(b) =(3a+2)*(3b+2)
Since both are not equal, C is Wrong.

D. f(a+b) = sqrt(2a+2b)
f(a)*f(b) = \sqrt{2a} +\sqrt{2b}
Since both are not equal, D is Wrong.

E. f(a+b) = 24^(a+b)
f(a)*f(b) = 24^(a+b)
Since both are equal, E is Correct.

(a) is (a+b)ˆ2+1=(aˆ2+1)(bˆ2+1)? no.
(b) is (5)ˆ2(a+b)/3=(5)ˆ2(a+b)/9? no.
(c) is 3(a+b)+2=(3(a)+2)(3(b)+2)? no.
(d) is √2(a+b)=2√ab? no.

Answer (E): 24ˆ(a+b)=24^a*24^b=24^(a+b)

Quickest way - put in values of a+b and a & b in the given functions

f(x)=x^2+1; f(a+b)=(a+b)^2+1 and f(a)=a^2+1 * f(b)=b^2+1 CLEARLY NO

f(x)=5^2x/3, now solving this will give you a clear hint that this is not the answer since the denominator on the right and side will be 3*3, which was a pretty clear indication for me to jump straight to option (E)

f(x)=24^x; f(a+b)=24^(a+b) and f(a)=24^a * f(b)=24^b, now since we know that when the base is same the powers add, this makes it the correct answer choice, and hence, (E) is our correct answer choice

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

A. f(x)=x2+1

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

C. f(x)=3x+2

D. f(x)=2x

E. f(x)=24^x

evaluating expressions should be of the form f(a+b) =f(a)*f(b)

Most are multiliplaction
generally powers have a rule of x^a *x^b = x^(a+b)
i saw E as first such option
and put in the value
\(24^a*24^y = 24^(a+b)\)
Which fulfils the given expression

Hence answer is E
_________________

A. \((a+b)^2+1=a^2+b^2+2ab+\)1 which is not necessarily equals to \((a^2+1)*(b^2+1)\)
B. \(\frac{5^{2a+2b}}{3}\)= \(\frac{5^{2a}}{3}\)*\(5^{2b}\), not equal to \(\frac{5^{2a}}{3}\)* \(\frac{5^{2b}}{3}\)

C. 3(a+b)+2=3a+3b+2 is not equal to (3a+2)*(3b+2)
D. \(\sqrt{2(a+b)}\) is not equal to \(\sqrt{2a}*\sqrt{2b}\)
E. \(24^{a+b}=24^a*24^b\)

IMO E

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)=a^2*b^2+a^2+b^2+1
f(a+b) doesn't equal f(a)*f(b)

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^(2a+2b)/9
f(a+b) doesn't equal f(a)*f(b)

C. f(x)=3x+2
f(a+b)=3(a+b)+2
f(a)f(b) = (3a+2)(3b+2)
f(a+b) doesn't equal f(a)*f(b)

D. f(x)=(2x)^(1/2)
f(a+b)=(2(a+b))^(1/2)
f(a)f(b) = (2a)^1/2 * (2b)^1/2 = 2(ab)^1/2
f(a+b) doesn't equal f(a)*f(b)

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

Answer: E

IMO E

A. f(x)=x^2+1 => f(a+b) = a^2+b^2+2ab+1; f(a)*f(b) = (a^2+1)(b^2+1) = a^2*b^2+a^2+b^2+1

B. f(x)=(5^(2x))/3 = (25^x)/3 => f(a+b) = (25^(a+b))/3 = (25^a*25^b)/3; f(a)*f(b) = ((25^a)/3)*((25^b)/3) = (25^a*25^b)/9

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

D. f(x)=√(2x) => f(a+b) = √(2a+2b); f(a)*f(b) = √(2a)*√(2b) = 2√(ab)

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

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

I found the easiest way to solve this by plugging values - let a=2 and b=3
So we will check whether f(2)*(f*3) = f(5)

There may be a better way to solve it

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

f(2)=5
f(3)=10
f(5)=26
f(2)*f(3)=50

INCORRECT


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

f(2)= 5^4/3
f(3)=5^6/3
f(5)=5^10/3
f(2)*f(3)=5^10/9

INCORRECT

C. f(x)=3x+2

f(2)=8
f(3)=11
f(5)=17
f(2)*f(3)=88

INCORRECT

D. f(x)=√2x

f(2)=√4
f(3)=√6
f(5)=√10
f(2)*f(3)=√24

INCORRECT

E. f(x)=24^x

f(2)=24^2
f(3)=24^3
f(5)=24^5
f(2)*f(3)=24^5

CORRECT

Answer: E

\(f(a+b)=f(a)∗f(b)\) means that \(f(5+5) = f(5) * f(5)\) or \(f(7+3) = f(7) * f(3)\) or etc.

Let's check answer choices by plugging in numbers:

A. Is \(f(x)=x^2+1\) true for \(f(5+5) = f(5) * f(5)\) ?

\(f(5+5) = f(10) = 10^2+1 = 101\)

\(f(5) = 5^2+1 = 26\)

So the result of \(f(5+5) = f(5) * f(5)\) would be \({101}\neq{26+26}\). A is out.

Plugging the same numbers in B, C, and D, we can see that they are incorrect.

E. Is \(f(x)=24^x\) true for \(f(5+5) = f(5) * f(5)\) ?

\(f(5+5) = f(10) = 24^{10}\)

\(f(5) = 24^5\)

So the result of \(f(5+5) = f(5) * f(5)\) would be \(24^{10} = 24^5*24^5\). Correct

Hence E
_________________

The correct answer is E, because

24^(a+b)=(24^a)*(24^b)
24^(a+b)= 24^(a+b)

Answer is E.
A=2; b=3; a+b=5

e) f(x)=24x
24ˆ5= 24ˆ2+24ˆ3=2+3=5

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.

by just looking at equations and substituiting
2and 3 for a& b

24^5=24^2*24^3=24^(2+3)

option E