Problem 91 - Algebra and Trigonometry

Question

When Does a Graph Represent a Function? For every integer \(n\) , the graph of the equation \(y=x^{n}\) is the graph of a function, namely \(f(x)=x^{n} .\) Explain why the graph of \(x=y^{2}\) is not the graph of a function of \(x\) . If so, of what function \(x=y^{3}\) the graph of a function of \(x ?\) If so, of what function of \(x\) is it the graph? Determine for what integers \(n\) the graph of \(x=y^{n}\) is the graph of a function of \(x .\)

Step-by-Step Solution

Answer

The graph \(x = y^n\) is a function of \(x\) for odd integers \(n\).

Step 1: Understand Function Definition

A function is a relation in which each input (often represented as \(x\)) has exactly one output (often represented as \(y\)). This means for every \(x\), there is one and only one corresponding \(y\) value.

Step 2: Analyze Equation \(x = y^2\)

Rewrite the equation as \(y = \pm \sqrt{x}\). This indicates that for each positive \(x\) value, there are two corresponding \(y\) values (one positive and one negative), which violates the definition of a function. Hence, \(x = y^2\) does not represent a function of \(x\).

Step 3: Analyze Equation \(x = y^3\)

We can express \(y\) as \(y = x^{1/3}\), for every \(x\) value there is exactly one corresponding \(y\) value, hence \(x = y^3\) represents a function of \(x\).

Step 4: Generalize for Equation \(x = y^n\)

Consider the equation \(x = y^n\). If \(n\) is even, \(x\) will be positive for two possible \(y\) values unless \(y = 0\). If \(n\) is odd, then for every \(x\) there will be exactly one possible \(y\). Hence, \(x = y^n\) is a function of \(x\) if \(n\) is odd.

Key Concepts

Function Definition

A function is a fundamental concept in mathematics. It serves as a bridge between two sets, linking each element from the first set, the domain, to one and only one element in the second set, the range. In simple terms, a function

• Assigns a single output to each input, ensuring no input leads to multiple outputs.
• Can be thought of as a machine that operates only one way with predictable outcomes.

For example, the simple equation equation implies that if you input any value for \(x\) into the function \(f(x) = x^n\), you will receive just one corresponding result for \(y\). As a visual cue, imagine a function as a vending machine: you press one button (your input), and you get one snack – never more. This holds the uniqueness of each pair in a function, making mathematical processes predictable and consistent.

Even and Odd Functions

Even and odd functions are specific types of functions with distinct symmetry properties. Understanding these helps us determine function characteristics and predict their behavior in relation to their graphs. - **Even Functions** exhibit symmetry around the y-axis. This means if you reflect the graph across the y-axis, it looks identical. - Mathematically, even functions satisfy the condition \(f(-x) = f(x)\) for all \(x\) in the domain. An example of this symmetry can be seen in the function \(x = y^2\), though it is identically reflected, it does not define a proper function of \(x\) due to multiple \(y\) values for a single \(x\).- **Odd Functions** show symmetry around the origin. Rotate an odd function’s graph 180 degrees, and it appears unchanged. - **Formal Definition** of odd functions is \(f(-x) = -f(x)\) for all \(x\) in the domain. The equation \(y = x^{1/3}\) is a classic example, treating negative inputs similarly due to its odd index. Knowing whether a function is even or odd can aid in solving equations and understanding the graph's behavior over the coordinate plane. They also help in simplifying integrals and identifying function constraints.

Vertical Line Test

The vertical line test is a quick and intuitive manual process for determining if a curve in the coordinate plane is the graph of a function.- **How It Works:** Imagine drawing vertical lines (lines parallel to the y-axis) across your graph. If any of these lines intersect the curve at more than one point, the graph is not of a function.This test is based on the definition that functions attribute only one output to each input - meaning, for each \(x\) value on a graph, there should be only one \(y\) value. Therefore:

• For a graph to represent a function, every vertical line should touch it at most once.
• For example, with the equation \(x = y^2\), drawing vertical lines shows they cross the curve at two points, revealing it's not a function of \(x\).
• Conversely, the graph of \(x = y^3\) satisfies the test, as each vertical line touches it just once, confirming a function of \(x\).

Using this test is an efficient way to visually validate functions, ensuring they meet mathematical criteria before deeper analysis or application.