Problem 13 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. $$ g(x)=\sqrt{x} $$

Step-by-Step Solution

Answer

The function \( g(x) = \sqrt{x} \) is one-to-one because different inputs yield different outputs.

Step 1: Understand the Definition of One-to-One Function

A function is one-to-one (injective) if for every pair of different inputs, the outputs are also different. In other words, if \( g(a) = g(b) \) implies \( a = b \), then the function is one-to-one.

Step 2: Check if the Function g(x) = √x Is One-to-One

Consider two different inputs \( a \) and \( b \) such that \( g(a) = g(b) \), which means \( \sqrt{a} = \sqrt{b} \). Since taking the square root of both sides implies \( a = b \), the function is one-to-one as it satisfies the condition \( g(a) = g(b) \) only when \( a = b \).

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, has a unique quality. Each output value is paired with no more than one input value. This means that if you have two different inputs, say \( a \) and \( b \), and they produce the same output, the only way this can happen is if \( a \) and \( b \) are actually the same number.

• To test if a function is one-to-one, check if different inputs yield different outputs.
• The formal way to define it is: A function \( f(x) \) is one-to-one if \( f(a) = f(b) \) implies that \( a = b \).

This property ensures that the function has an inverse since each output maps back to one and only one input. A quick way to verify if a function is injective is through the "Horizontal Line Test": If no horizontal line intersects the graph more than once, the function is one-to-one.

Function Properties

Functions have a variety of properties that define their behavior. Understanding these properties helps us decide the nature of the function and how to handle it in math problems.

• Domain and Range: This tells us what values a function can accept as inputs (domain) and what values it can output (range).
• Continuity: A continuous function has no breaks, jumps, or holes in its graph.
• Monotonicity: A function is called monotonic if it is either entirely non-increasing or non-decreasing.

Knowing these properties helps us understand functions better. They tell us how functions behave, which is essential for solving equations or analyzing data trends.

Square Root Function

The square root function is popular in mathematics, denoted as \( g(x) = \sqrt{x} \). It is a type of radical function that specifically deals with finding the square root of its input. Here’s what you need to know about it:

• Domain: The inputs for a square root function are non-negative numbers — meaning \( x \geq 0 \). This is because you can't take the square root of a negative number without involving complex numbers.
• Graph: The graph starts from the origin \((0,0)\) and rises slowly, forming a curve that increases at a decreasing rate.
• One-to-One: As shown earlier, the square root function is one-to-one because different inputs produce different outputs, fulfilling the requirement for injectivity.

In practical terms, square root functions appear in various applications, such as calculating the side length of a square, given its area. They are also fundamental in solving quadratic equations.