Problem 84 - Algebra and Trigonometry

Question

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 \(x .\) Is the graph of \(x=y^{3}\) 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 of \(x = y^n\) is a function of \(x\) for odd \(n\).

Step 1: Define the function

A graph represents a function if every input value (x-value) is paired with exactly one output value (y-value). In other words, no single x-value should correspond to more than one y-value.

Step 2: Analyze x = y^2

For the equation \(x = y^2\), if we attempt to solve for \(y\), we get \(y = \pm\sqrt{x}\). This gives two possible y-values for each positive x-value, meaning this equation is not a function of \(x\).

Step 3: Analyze x = y^3

For the equation \(x = y^3\), solving for \(y\) yields \(y = x^{1/3}\). Since each x-value results in exactly one y-value, this is a function of \(x\).

Step 4: Generalize for x = y^n

Consider \(x = y^n\). If \(n\) is odd, solving for \(y\) gives \(y = x^{1/n}\), yielding one unique y-value for each x-value. If \(n\) is even, \(y = \pm x^{1/n}\), leading to two y-values for positive x-values, which means it is not a function. Therefore, \(x = y^n\) is a function of \(x\) for odd \(n\) only.

Key Concepts

Graph of a function

A graph of a function visually represents the relationship between inputs (usually x-values) and their corresponding outputs (y-values). If you can draw a vertical line through any point on the graph and it touches the graph at most once, it's a graph of a function. This is known as the vertical line test.

For a function, every input value has exactly one output. For example, the graph of the function \( y = x^n \) is valid because each x-value gives a specific y-value. Disallowing any input from having more than one output ensures clarity and correctness in representing data.

• Single x-value to y-value correspondence
• Passes the vertical line test
• Functional representation

Understanding this helps to distinguish functions from non-functions.

Integer exponents

Integer exponents reveal how many times to multiply the base by itself. For example, \( x^n \) means multiply \( x \) by itself \( n \) times. This concept is crucial when dealing with functions like \( y = x^n \), where the graph's shape depends on whether \( n \) is positive, negative, odd, or even.

When n is:

• Positive: The exponential relationship grows with increasing values. For example, \( x^2 \) results in a parabolic upward curve.
• Negative: The relationship inverts, creating curves that approach the axis asymptotically. For example, \( x^{-1} \) forms a hyperbola.
• Zero: Any number raised to the power of zero equals 1, making \( x^0 = 1 \).

Recognizing these differences aids in graphing and comprehending behavior patterns in mathematical functions.

Odd and even functions

Odd and even functions display distinctive symmetries in their graphs.

An odd function is symmetric about the origin, implying \( f(-x) = -f(x) \). Graphically, this means rotating the graph 180 degrees around the origin results in the same graph. Cubic functions like \( f(x) = x^3 \) are classic examples.

• Symmetrical about origin
• Graph appears consistent upon rotation

In contrast, an even function is symmetric about the y-axis, meaning \( f(x) = f(-x) \). An example is \( f(x) = x^2 \), where the graph mirrors itself across the y-axis.

• Symmetrical about the y-axis
• Graph identical on either side of the y-axis

Recognizing whether a function is odd or even can simplify the process of graphing and understanding its properties.

Cube roots

Cube roots help in understanding the inverse operations of cube functions, like \( x = y^3 \). Solving such an equation involves finding \( y \) such that \( y = x^{1/3} \), or the cube root of \( x \). Unlike square roots, cube roots yield one real solution for both positive and negative inputs, making them particularly useful in graphing functions of the form \( y = x^{1/3} \).

Key insights about cube roots include:

• Single solution: Each x-value corresponds to exactly one y-value, ensuring it functions mathematically.
• Handles negatives: Cube roots can compute real results from negative x-values, unlike square roots which necessitate complex numbers.

Understanding how cube roots work clarifies why graphs of cube functions represent distinct functions and maintains essential properties across all real numbers.