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.\nanswer\n( f(x) ) has an even degree, but is not an even function\n( f(x) ) has an even degree and is an even function\n( f(x) ) has an odd degree, but is not an odd function\n( f(x) ) has an odd degree and is an odd function\nsubmit answer
Answer
Explanation:
Step1: Check the symmetry of the function
An even function satisfies (f(x)=f(-x)) (symmetric about the y - axis), and an odd function satisfies (f(-x)=-f(x)) (symmetric about the origin). Looking at the graph, when (x) is replaced with (-x), the graph is symmetric about the origin. For example, if we take a point ((a,b)) on the graph, the point ((-a, -b)) is also on the graph.
Step2: Determine the end - behavior to check the degree
For a polynomial function (y = f(x)=a_nx^n+\cdots+a_1x + a_0), if (n) is even, as (x\to\pm\infty), (y) has the same sign (both tend to (+\infty) if (a_n>0) or both tend to (-\infty) if (a_n < 0)). If (n) is odd, as (x\to+\infty) and (x\to-\infty), (y) has opposite signs. In the given graph, as (x\to+\infty), (y\to+\infty) and as (x\to-\infty), (y\to-\infty), which is the behavior of a function with an odd degree.
Answer:
(f(x)) has an odd degree and is an odd function.