the function $f(x)$ is graphed below. determine whether the degree of the function is even or odd and…

the function $f(x)$ is graphed below. determine whether the degree of the function is even or odd and whether the function itself is even or odd.

the function $f(x)$ is graphed below. determine whether the degree of the function is even or odd and whether the function itself is even or odd.

Answer

Explanation:

Step1: Analyze End Behavior for Degree

The end behavior of a polynomial function is determined by its leading term. For the given graph, both ends of the graph (as ( x \to \infty ) and ( x \to -\infty )) are rising (going up). For a polynomial ( f(x)=a_nx^n + \dots+a_0 ), if the leading coefficient ( a_n>0 ) and the degree ( n ) is even, the end behavior is ( f(x)\to\infty ) as ( x\to\pm\infty ). If ( n ) is odd, the ends go in opposite directions (one up, one down). Since both ends are up, the degree ( n ) is even.

Step2: Check Symmetry for Even/Odd Function

A function is even if ( f(-x)=f(x) ) (symmetric about the ( y )-axis) and odd if ( f(-x)= - f(x) ) (symmetric about the origin). The graph shown is symmetric about the ( y )-axis (the left side is a mirror image of the right side across the ( y )-axis). So, by the definition of an even function, ( f(-x)=f(x) ), so the function is even.

Answer:

The degree of the function is even, and the function itself is even.