a A function f is an even function when f(- x)=f(x) for all x in its domain.
B
b A function f is an odd function when f(- x)=- f(x) for all x in its domain.
C
c Analyze the answers from Parts A and B.
A
a b=0, and the values of a and c are arbitrary.
B
b a=0, c=0, and the value of b is arbitrary.
C
c Neither b=0 nor a=0, and c=0.
Practice makes perfect
a Before we begin, let's recall the definition of an even function.
A function f is an even function if and only if f(- x)=f(x) for all x in its domain.
We are asked to find values of a, b, and c, so that the function f(x)=ax^2+bx+c is even. Let's simplify the equation f(- x)=f(x) in the case of the given function.
Since equation bx=0 should be valid for all values of x, we get that b=0.
b Let's recall the definition of an odd function.
A function f is an odd function if and only if f(- x)=- f(x) for all x in its domain.
We are asked to find values of a, b, and c, so that the function f(x)=ax^2+bx+c is even. Let's simplify the equation f(- x)=- f(x) in the case of the given function.
Since equation ax^2+c=0 should be valid for all values of x, we get that a=0 and c=0.
c From Part A we know that the function f is even if and only if b=0. From Part B we know that the function f is odd if and only if a=0 and c=0. Therefore, the function f is neither even nor odd if and only if values of a, b, and c are neither b=0 nor a=0 and c=0.
