**Ursinus MATH 111: Calculus 1**

## Week 1 Challenge Problems

### Christopher Tralie

## The Sum of Even Functions

###
__Question:__

If `f`

is even and _{1}(x)`f`

is even, then _{2}(x)`f(x) = f`

is
_{1}(x) + f_{2}(x)

- Even
- Odd
- Not necessarily either

__Answer__

First, it might be a good idea to start playing around with functions. For example, what if you let \[f_1(x) = 5x^2\] \[f_2(x) = -x^4 \]. Let's look at this maple (you can also test values by hand)

`f`

certainly looks even in this example! And you can try other examples and you will continue to see even results. But how do we prove this in general? Well, let's go back to the definition of even functions: a function `g`

is even if and only if \[ g(x) = g(-x) \]
Using this fact, we can do the following transformation to `f(x)`

:

\[ f(-x) = f_1(-x) + f_2(-x) \] | Plugging in `-x` |

\[ f(-x) = f_1(x) + f_2(x) = f(x) \] | Using the fact that `f1` and `f2` are even functions |

`f(x)`

**is even**

## The Sum of Odd Functions

###
__Question:__

If `f`

is odd and _{1}(x)`f`

is odd, then _{2}(x)`f(x) = f`

is
_{1}(x) + f_{2}(x)

- Even
- Odd
- Not necessarily either

__Answer__

Again, let's play around with some odd functions. For example, what if you let \[f_1(x) = x^3 \] \[f_2(x) = -2x \]. Let's look at this maple (you can also test values by hand)

`f`

certainly looks odd in this example! And you can try other examples and you will continue to see odd results. But how do we prove this one now? Again, let's apply the definition: a function `g`

is odd if and only if \[ g(x) = -g(-x) \]
Using this fact, we can do the following transformation to `f(x)`

:

\[ f(-x) = f_1(-x) + f_2(-x) \] | Plugging in `-x` |

\[ f(-x) = -f_1(x) + -f_2(x) = -f(x) \] | Using the fact that `f1` and `f2` are odd functions and factoring out a negative |

`f(x)`

is odd
## The Sum of An Even And Odd Function

###
__Question:__

If `f`

is odd and _{1}(x)`f`

is even, then _{2}(x)`f(x) = f`

is
_{1}(x) + f_{2}(x)

- Even
- Odd
- Not necessarily either

__Answer__

Again, let's play around with some odd functions. For example, what if you let \[f_1(x) = x^3 \] \[f_2(x) = x^2 \]. In this example, \[ f(-1) = (-1)^3 + (-1)^2 = 0\] and \[ f_1(1) = (1)^3 + (1)^2 = 2 \] Uh oh. Actually, we've already found a counter example where the function is neither even or odd. So in general, all bets are off with the sum of even and odd functions