Problem 80 - College Algebra with Modeling and Visualization

Question

Is there an even function whose domain is all real numbers and that is always decreasing? Explain.

Step-by-Step Solution

Answer

No, a non-trivial even function that is always decreasing on all real numbers does not exist.

Step 1: Understanding Even Function

An even function is defined as a function where for every input \(x\), the corresponding output is the same as for \(-x\). Mathematically, this means \(f(x) = f(-x)\) for all \(x\) in its domain.

Step 2: Properties of Decreasing Function

A decreasing function is one where, for any two values \(x_1\) and \(x_2\) in its domain, if \(x_1 < x_2\), then \(f(x_1) \geq f(x_2)\). That means the function's output decreases or stays the same as the input increases.

Step 3: Attempting to Reconcile Properties

For an even function to be always decreasing, it would have to be decreasing in both positive and negative directions simultaneously. However, if \(figg( \frac{x}{2} \bigg) \geq f(x) \geq figg( \frac{x}{2} \bigg)\), then it would imply \(f(x) = figg( \frac{x}{2} \bigg)\) constantly decreasing, suggesting a constant function.

Step 4: Constant Functions Investigation

A constant function is the only function that can be considered both even and everyday decreasing across all domains. However, a constant function is also not decreasing in the typical sense as it maintains the same value across its domain rather than reducing.

Step 5: Conclusion

Thus, there is no non-trivial function that is even, has a domain of all real numbers, and is always decreasing. The only potential candidate, a constant function, does not strictly decrease.

Key Concepts

Decreasing Function

Imagine a hill where you always step downward as you walk. This is similar to how a decreasing function behaves. In math terms, a function is decreasing if, whenever you pick two numbers where the first one is smaller than the second (say, \(x_1 < x_2\)), the function's value at the first number is greater than or equal to its value at the second (so, \(f(x_1) \geq f(x_2)\)).
This means that as the input numbers get bigger, the output values get smaller or stay the same.

• Tip: Think about sliding down a slide. You always move down or stay flat, never up.
• If it is strict, it always drops without staying the same; if not, it can be flat for some intervals.

Constant Function

Picture a very straight path that never turns or dips. A constant function is exactly like that. For all \(x\) in its domain, the output never changes. It's like a flat road stretching infinitely.
In more formal terms, for a function to be constant, \(f(x) = C\) where \(C\) is the same for any \(x\).

• Example: No matter how far you go down this path, you're always at the same level.
• Tip: Imagine an idle see-saw that never tips. Both sides stay level always.

This function is also special because it is the only function that can be considered both even and decreasing, though it feels more "flat" than sloping downward.

Real Numbers Domain

When we talk about a domain of real numbers, imagine using every possible number you can think of. Real numbers include everything: whole numbers, decimals, fractions, and even nasty, unpredictable irrational numbers like \(\pi\) and \(e\).

• This domain is vast and continuous, like an endless ocean of numbers.
• Tip: Whenever you see a domain like this, realize there are no gaps or breaks in the numbers you can use.

Functions that can handle all real numbers are powerful. They have no restrictions as long as the math checks out, meaning they can work continuously across any point on the number line.

Function Properties

Functions are like magical machines that take inputs (numbers) and transform them into outputs. They can stretch, flip, and morph in many ways, and certain properties help us understand their behavior. Here are a few key function properties that can help in navigating and analyzing functions:

• Even Function: If \(f(x) = f(-x)\), the function is even, meaning it is symmetrical about the vertical axis.
• Odd Function: If \(f(-x) = -f(x)\), the function is odd, which may look like a kind of diagonal symmetry.
• Increasing Function: When the output gets larger as you input larger numbers.
• Decreasing Function: Opposite of increasing, discussed earlier!

Understanding these properties can simplify solving equations and predicting how functions will behave over different intervals.