Question

Which of the following is true regarding the symmetry of the function ? \[f(x) = x^{4}\ + x^{2}\ + 3\]
A. \[f(x)\ = f( - x)\]
B. \[f(x) = - f( - x)\]
C. It is an even function.
D. Symmetric wrt y axis

Answer 1

Hint: In this question, we need to know which of the given statements is true regarding the symmetry of the given function. First, let us know the concept of symmetry . That is if the given function is even then it is symmetric about the y axis and similarly if the given function is odd then it is symmetric about the x axis. First we need to find whether the given function is even or odd. For that we need to substitute \[- x\] in the place of \[x\].

Complete step-by-step answer:
Given function \[f(x) = x^{4}\ + x^{2}\ + 3\]
First we need to find whether the given function is even or odd.
Now we can substitute \[- x\] in the place of \[x\].
On substituting,
We get,
\[\Rightarrow \ f( - x)\ = ( - x)^{4} + ( - x)^{2} + 3\]
On simplifying,
We get,
\[f( - x) = x^{4}\ + x^{2}\ + 3\]
Hence we can tell that \[f(x)\] is equal to \[f( - x)\]
Hence the given function is even.
According to the concept of symmetric, if the given function is even then it is symmetric about y axis
Thus the given function is symmetric about the y axis.

Thus the true statement is \[f(x)\ = f( - x)\] , It is an even function and Symmetric wrt y axis.
Final answer :
The true statement is \[f(x)\ = f( - x)\] , It is an even function and Symmetric wrt y axis.
Option A, C, D are the true statements.

So, the correct answer is “Option A,C and D”.

Note: In order to solve these types of questions, we should have a strong grip over even and odd functions. If \[f(x)\] is equal to \[f( - x)\], then the function is said to be the even function similarly if \[f(x)\] is equal to \[- f( - x)\] then the function is said to be the odd function. We also need to know that if a function is both even and odd, then it is equal to \[0\] everywhere it is defined similarly if a function is odd, the absolute value of that function is an even function