Question

Monotonic Functions Show that monotonic increasing and decreasing functions are one-to-one.

Short answer

A monotonic function is one-to-one if for any two different points in its domain, no two have the same image. This is intrinsic to the property of being monotonically increasing or decreasing.

Step-by-step solution

Case 1 Monotonically Increasing

Assume a function \( f(x) \) is monotonically increasing in its domain. If \( x_1 \) and \( x_2 \) are two different points in the domain and \( x_1 > x_2 \) then by the properties of a monotonically increasing function \( f(x_1) > f(x_2) \). Hence, no two different points in the domain have the same image and the function \( f(x) \) is one-to-one.

Case 2 Monotonically Decreasing

Assume a function \( f(x) \) is monotonically decreasing in its domain. If \( x_1 \) and \( x_2 \) are two different points in the domain and \( x_1 > x_2 \) then by the properties of a monotonically decreasing function \( f(x_1) < f(x_2) \). Hence, no two different points in the domain have the same image and the function \( f(x) \) is one-to-one.

Key concepts

One-to-One Functions

Imagine you are at a party, and every guest is given a unique gift. Just like these gifts, in mathematics, a function is called a one-to-one function (or injective) if it gives a different output for every different input. It's like having a special rule that ensures no one at the party gets the same gift.

For every unique value you put into this function, you're guaranteed to get a unique result back out. Why does this matter? Such functions make it possible to talk about 'reversing' the process to get back the original input, leading to the concept of 'inverses' in math.

Monotonically Increasing Functions

Let's go on a hike, but on a path that only goes up, never down. This is similar to how a monotonically increasing function behaves: for any two different points you choose along its journey, the later one is always higher or at least equal in elevation compared to the earlier one. Mathematically speaking, if you have two inputs, and the second is larger, the output of the second will be greater than or equal to that of the first.

This consistent upward trend guarantees that each input has a distinct output, leading back to our concept of one-to-one functions.

Monotonically Decreasing Functions

Imagine now your hike is on a slope heading strictly downwards. This captures the essence of a monotonically decreasing function: as you move along with the inputs, the outputs fall or remain constant. If you select any two different points where the second is past the first, the result for the second point will always be less than or equal to that of the first.

This never-rising characteristic is what secures the function's status as one-to-one. No matter how many steps you take downhill, you will never end up at a higher point than where you started.

Function Properties

So, what makes a function special? Functions come with a toolkit of properties that describe their behaviors, like the directions they move (increasing or decreasing) and how they pair inputs to outputs (one-to-one).

When a function is both one-to-one and either monotonically increasing or decreasing, we have a clear understanding of how it behaves across its entire domain. These properties help us to analyze the functions, predict outcomes, and even find inverses. They also ensure the function passes what's known as the 'Horizontal Line Test'—a useful visual aid in determining if a function is one-to-one simply by drawing horizontal lines through its graph.