Question

A student says that the function f(x)=3x4+5x2+1 is an even function.Is the student's statement true or not true, and why?

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given function

`f(x)=3x^4+5x^2+1`

STEP 2: State when a function is said to be an even function

A function is an even function if f(x) is equal to f (−x) for all the values of x. This means that the function is the same for the positive x-axis and the negative x-axis, or graphically, symmetric about the y-axis.

STEP 3: Find if the given function is even

Substitute -x in place of x in f(x)

`\begin{gathered} f(x)=3x^(4)+5x^(2)+1 \\ f(-x)=3(-x^4)+5(-x^2)+1 \\ =3(-x\cdot-x\cdot-x\cdot-x)+5(-x\cdot-x)+1 \\ =3(x^4)+5(x^2)+1 \\ =3x^4+5x^2+1=f(x) \end{gathered}`

ANSWER:

The student's claim is true, because for any input of x, f(x) = f(-x)

OPTION A