Thread: Odd-Even Functions & Polynomial Functions

1. Odd-Even Functions & Polynomial Functions

1. Let f(x) be a function about which we know nothing. Find which of the following statements is false.

A. g(x) = f(x) + f(-x) is an even function
B. g(x) = f(x) - f(-x) is an odd function
C. g(x) = f(|x|) is an even function
D. g(x) = f(-|x|) is an odd function
E.. g(x) = f(x^2) is an even function

2. Assume that f(x) and g(x) are both polynomial functions. Which one of the following statements is necessarily true?

A. If the degree of f(x) is bigger than the degree of g(x), then f(x) has more distinct solutions than g(x).
B. If the degree of f(x) is bigger than the degree of g(x), then f(x) has more distinct solutions (counting multiplicities) than g(x).
C. If the degree of f(x) is even, and the degree of g(x) is odd, then their graphs are going to cross at least once.
D. If the degrees of both f(x) and g(x) are even, then the degree of f(x) + g(x) is also even.
E. If the degrees of both f(x) and g(x) are even, then the degree of f(x)g(x) is also even.

3. Assume that f(x) is a polynomial function of degree 3, and the leading coefficient is 1. Assume moreover, that f(x) has 2 distinct solutions. Then:

A. No such function is possible.
B. There is a unique such function.
C. There are 2 different such functions.
D. There are 3 different such functions.
E. There are 4 different such functions.

2. Originally Posted by tetrachordfetuses
1. Let f(x) be a function about which we know nothing. Find which of the following statements is false.

A. g(x) = f(x) + f(-x) is an even function
B. g(x) = f(x) - f(-x) is an odd function
C. g(x) = f(|x|) is an even function
D. g(x) = f(-|x|) is an odd function
E.. g(x) = f(x^2) is an even function
An odd function is a function $h(x)$ , such that:

$h(-x)=-h(x)$

and an even function is a function $h(x)$, such that:

$h(-x)=h(x).$

Now I will do one of these as an example:

"A. $g(x) = f(x) + f(-x)$ is an even function"

so:

$g(-x)=f(-x)+f(x)=f(x)+f(-x)=g(x)$

and we conclude that $g$ is an even function.

CB

3. [quote=tetrachordfetuses;210710]
2. Assume that f(x) and g(x) are both polynomial functions. Which one of the following statements is necessarily true?

A. If the degree of f(x) is bigger than the degree of g(x), then f(x) has more distinct solutions than g(x).
B. If the degree of f(x) is bigger than the degree of g(x), then f(x) has more distinct solutions (counting multiplicities) than g(x).
C. If the degree of f(x) is even, and the degree of g(x) is odd, then their graphs are going to cross at least once.
D. If the degrees of both f(x) and g(x) are even, then the degree of f(x) + g(x) is also even.
E. If the degrees of both f(x) and g(x) are even, then the degree of f(x)g(x) is also even.

A. If $f(x)$ is a quadratic with no real roots (I presume we are talking real solutioms here) and $g(x)=a.x$ Then $f$ and $g$ satisfy the conditions of the premis but $g$ has more distinct solutions that $f$.

B. The counterexample given above to A. is also a counter example for this

C. Sketch a few quadratics and linear functions and see if you can answer this.

D. Consider $f(x)=x^2+x$, $g(x)=-x^2$

E. Consider the degree of the highest degree term in the product of two even degree polynomials.

CB