Question

Let f be an even periodic function with period 2 and $f\left(x\right)=x\mathrm{\forall }x\in \left[0,1\right]$.Then $f\left(3.14\right)$ equals
A.-3.14
B.-0.14
C. 0.14
D. 0.86

Answer 1

Hint: We know that if a function is even then will be positive that is $f\left(-x\right)=f\left(x\right)$ for all values of x, which means that the function is symmetric about the y-axis for all the values of x and also the given function is periodic which means that it has same values after a certain period. So, using these properties we are going to solve this question.


Complete step by step solution
Given:
The function is $f\left(x\right)=x\mathrm{\forall }x\in \left[0,1\right]$.
Since we know the period of the function f is given as 2, that is,
$f\left(x\right)=f\left(x+2\right)$
It means that if we put $x=0$ in the function then the value of $f\left(0\right)$ would be the same as that of the value of the $f\left(2\right)$.
So also, we can say that $f\left(x\right)=f\left(x+2\right)=f\left(x+4\right)=x$.
Now, to calculate the value of the function at 3.14, we can write $f\left(3.14\right)$ in a following way,
$\begin{array}{c}f\left(3.14\right)=f\left(4-0.86\right)\\ =f\left[4+\left(-0.86\right)\right]\end{array}$
On comparing this with $f\left(x+4\right)=x$ we get,
$f\left(3.14\right)=f\left(-0.86\right)$
Since, we know the function is even that is $f\left(-x\right)=f\left(x\right)$ we get,
$f\left(3.14\right)=f\left(0.86\right)$
Also, we know that for this function $$f\left(x\right)=x$$, so we get,
$f\left(3.14\right)=0.86$
Therefore, the correct option is (D) that is $0.86$.


Note: Always remember that for a periodic function the value of the function repeats itself after a certain period. For example if a function $f\left(x\right)$ is represented by $f\left(x+t\right)$ then $t$ is known as the period of the function and these types of functions are called periodic functions. The most common examples of these kinds of functions are trigonometric functions such as Sine and Cosine.