Question

If the index of any power function is zero, then the value of that function is
(a) 0
(b) 1
(c) -1
(d) none of these

Answer 1

Hint: According to the question, it is given that the index of the power function is zero. Let us assume a power function \[f(x)={{x}^{n}}\] such that x can be any real number.

Complete step-by-step solution -
\[f(x)={{x}^{n}},x\in R\] ………………(1)
If n is an even number, that is n is divisible by 2 then the assumed function \[f(x)={{x}^{n}}\] , is an even function.
Similarly, if n is an odd number, that is n is not divisible by 2 then the assumed function \[f(x)={{x}^{n}}\] , is an odd function.
In the question, it is given that the index of the power function is zero. It means that in the function \[f(x)={{x}^{n}}\] , we have the value of n equal to zero.
Putting n=0 in equation (1), we get
\[f(x)={{x}^{n}}={{x}^{0}}\] …………………..(2)
We know the property that any real number having 0 as its exponent is equal to 1.
\[f(x)={{x}^{n}}={{x}^{0}}\]
Here, x is any real number. Now, using this property, we have
\[f(x)={{x}^{n}}={{x}^{0}}=1\]
Therefore, the value of any power function having zero as its index is 1.
Hence, the correct option is (B).

Note: In this question, one can make a silly mistake in the property that any number having 0 as its exponent is equal to 0. Using this property one can replace the number \[{{x}^{0}}\] by 0 in the expression \[f(x)={{x}^{n}}\] . This is wrong. Therefore, we have to keep the correct property in our mind.