Problem 128 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It A function \(f\) is increasing over its entire domain. Does \(f\) have an inverse function? Explain.

Step-by-Step Solution

Answer

Yes, the function \(f\) does have an inverse because it is increasing over its entire domain, which means it is a one-to-one (bijective) function.

Step 1: Defining the properties of a bijective function

A function \(f\) is said to be bijective (or one-to-one and onto), if each element of the codomain is mapped with exactly one element of the domain. In simple terms, there are no repeats in output, and every possible output is covered.

Step 2: Analyzing the given function

The problem mentions that the function \(f\) is increasing over its entire domain. This implies that as you move from left to right along the curve of the function, the function values are always increasing, i.e., for any two numbers \(x1\) and \(x2\) in the domain, if \(x1 < x2\), then \(f(x1) < f(x2)\).

Step 3: Deciding if function \(f\) has an inverse

Since the function \(f\) is always increasing, it means that the function will not repeat any value in the range. This means the function is one-to-one (bijective). As per the property of a bijective function, it can be inferred that the function has an inverse.

Key Concepts

Bijective Function

When we talk about bijective functions, we're addressing a key concept in mathematics. Think of a bijective function as a perfect matchmaker. It assigns each input a unique output and ensures all possible outputs are already paired. If you imagine the domain and codomain as two sets of people, a bijective function ensures everyone has one perfect dance partner.

• A function is one-to-one if different inputs produce different outputs.
• A function is onto if every possible output has been hit by some input.
• Bijective means both one-to-one and onto. Every element has a unique pair.

So why is a bijective function special? Because it passes the criteria to have an inverse! Only bijective functions possess this quality of having perfect backwards pairings, making each output traceable back to exactly one input.

Increasing Function

An increasing function is like climbing a hill. As you move forward, you either go up or stay on the same level, never decreasing. This means as your input \(x\) increases, the output \(f(x)\) doesn't fall.

• Formally, for any pair of values \(x_1 and x_2\), if \(x_1 < x_2\), then \(f(x_1) \leq f(x_2)\).
• Functions that always climb upwards, meaning \(f(x_1) < f(x_2)\), are strictly increasing.

Being increasing has significant implications for a function's one-to-one property. Why? Because two different points in the domain won't map to the same point in the codomain. There are no overlaps or repeats in range values.
Thus, increasing, especially strictly, makes it easier to ensure a function is one-to-one, paving the way for being bijective!

One-to-One Function

One-to-one functions ensure that each input maps to a distinctive output. It's like assigning a unique nickname to each friend in your circle. No two friends get the same nickname.

• For every pair of distinct inputs \(x_1, x_2\), \(f(x_1) eq f(x_2)\).

In simple terms, the vertical line test helps us check this property: if a vertical line intersects a function's graph more than once, it's not one-to-one.
The beauty of a one-to-one function is its part of the story in making a function bijective. Combine this with being onto, and boom! You've got a function primed and ready for an inverse.