Problem 32 - Precalculus : Building Concepts and Connections

Question

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=x+3$$

Step-by-Step Solution

Answer

The function \(f(x)=x+3\) is neither even nor odd.

Step 1: Substitute x for -x in the function

First, substitute \(x\) for \(-x\) in the given function. The function becomes \(f(-x) = -x + 3\)

Step 2: Compare f(x) and f(-x) for even function property

To test if the function is even, you compare \(f(x)\) and \(f(-x)\). If \(f(x) = f(-x)\), then the function is even. In this case, \(f(x) = x + 3\) and \(f(-x) = -x + 3\). It is clear that \(f(x) \neq f(-x)\). Therefore, the function is not even.

Step 3: Compare f(x) and -f(-x) for odd function property

To test if the function is odd, compare \(f(x)\) and \(-f(-x)\). If \(f(x) = -f(-x)\), then the function is odd. If you replace \(f(-x)\) with \(-f(x)\), get \(-(-x + 3)\) which simplifies to \(x - 3\). It is clear that \(f(x) \neq -f(-x)\). Therefore, the function is not odd.

Key Concepts

Even Function

An even function is a type of function with a special symmetry property. When we look at a graph of an even function, it appears mirrored around the y-axis. But how do we mathematically determine if a function is even?

The key is to use the definition: A function is even if, for every input \(x\), the following holds true:

• \(f(x) = f(-x)\)

To decide if the function \(f(x) = x + 3\) is even, we substitute \(x\) with \(-x\). As calculated before, \(f(-x) = -x + 3\).

We then compare \(f(x)\) and \(f(-x)\):

• If \(f(x) = f(-x)\), the function is even.
• If not, the function is not even.

In this example, \(x + 3\) does not equal \(-x + 3\), meaning \(f(x)\) is not equal to \(f(-x)\). Thus, \(f(x) = x + 3\) is not an even function.

Odd Function

Odd functions have their own unique symmetry. This symmetry is around the origin, meaning if you rotate the graph 180 degrees, it looks the same. Mathematically, a function is odd if, for every input \(x\), the relationship is:

• \(f(x) = -f(-x)\)

Let's test if \(f(x) = x + 3\) fits this description. First, calculate \(-f(-x)\) by taking the negative of \(f(-x)\): \(-(-x + 3)\), simplifying to \(x - 3\).

Now, compare \(f(x)\) with \(-f(-x)\):

• If \(f(x) = -f(-x)\), the function is odd.
• If not, the function is not odd.

Since \(x + 3\) is not equal to \(x - 3\), we conclude that \(f(x) = x + 3\) is not an odd function. Understanding whether a function is odd or not can help analyze its symmetry properties.

Function Properties

Every function has properties that determine its behavior and characteristics. These properties can include being even, odd, or neither, and are important for understanding a function's graph and its symmetry.

Functions that are neither even nor odd don't exhibit the symmetrical properties we associate with those types of functions. Instead, their symmetry depends on other characteristics or they may not have any apparent symmetry at all.

Let’s reflect on the given function \(f(x) = x + 3\):

• We found that it is neither even nor odd, meaning it doesn’t have a y-axis reflection nor an origin symmetry.
• Such functions are quite common and analyzing them requires a look at other features like their monotonicity or discontinuities.

Understanding the basic functional properties helps in sketching graphs and predicting the behavior of functions in various contexts, paving the way for advanced study into calculus and algebra.