Problem 16 - Algebra 2 and Trigonometry

Question

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=\frac{8}{x}\)

Step-by-Step Solution

Answer

a. Hyperbola; b. Not a direct variation; c. Yes, it's one-to-one.

Step 1: Identifying the Type of Function

The function given is \( y = \frac{8}{x} \). This is a rational function, as it can be expressed as a ratio of two polynomials. The numerator is a constant (8), and the denominator is a linear polynomial (\(x\)).

Step 2: Graphing the Function

To graph \( y = \frac{8}{x} \), note that it is a hyperbola centered around the origin. The function is undefined at \(x = 0\), creating a vertical asymptote at the y-axis. The graph will have two branches: one in the first quadrant (where both \(x\) and \(y\) are positive) and one in the third quadrant (where both are negative).

Step 3: Determine if the Function is a Direct Variation

Direct variation means the function is of the form \( y = kx \) for some constant \(k\). In this case, \( y = \frac{8}{x} \) is not a direct variation since it involves division rather than multiplication. Direct variation implies a linear relationship, which is not present here.

Step 4: Determine if the Function is One-to-One

A function is one-to-one if each \(y\) value is paired with exactly one unique \(x\) value. For function \( y = \frac{8}{x}\), for each \(x > 0\), there is a corresponding positive \(y\), and for each \(x < 0\), there is a corresponding negative \(y\). No two different \(x\) values result in the same \(y\), so the function is one-to-one.

Key Concepts

Graphing Functions

Graphing functions requires understanding the structure of the function being graphed. With rational functions like \( y = \frac{8}{x} \), the graph can take on forms like that of a hyperbola. Here are some key points to consider:

• The graph has two separate branches.
• There is a vertical asymptote at \( x = 0 \), meaning the graph gets infinitely close to the \( y \)-axis but never touches it.
• Similarly, as \( y \to 0 \), the graph approaches the \( x \)-axis, leading to a horizontal asymptote.

When graphing such a function, focus on these asymptotic behaviors and note that the graph exists in the first and third quadrants:

• For positive \( x \), the corresponding \( y \) is also positive, resulting in a graph in the first quadrant.
• For negative \( x \), the \( y \) is negative, placing the graph in the third quadrant.

Start by plotting key points, observing the asymptotic behaviors, and drawing smooth curves to connect these points. This approach ensures you accurately depict the hyperbolic nature of \( y = \frac{8}{x} \).

Direct Variation

Direct variation is a concept that captures a specific type of linear relationship between two variables. It is characterized by equations of the form \( y = kx \), where \( k \) is a constant. Let's break this concept down:

• Direct variation implies a proportional relationship - as one variable increases, the other does so at a constant rate.
• Graphically, this is represented by a straight line passing through the origin.

In the case of the function \( y = \frac{8}{x} \), it does not show a direct variation:

• Instead of a straight line, the function is a hyperbolic curve.
• There is division, not multiplication involved in its expression, which veers away from the definition of direct variation.

Understanding this distinction is crucial when identifying direct variations and recognizing why \( y = \frac{8}{x} \) does not fall under this category.

One-to-One Functions

One-to-one functions are an essential concept in mathematics, allowing us to determine if a function can be inverted reliably. A function is one-to-one if each input corresponds to a unique output:

• This means no two different \( x \) values will yield the same \( y \) value.
• Graphically, a horizontal line test can verify this - a line drawn horizontally should intersect the graph only once.

For the function \( y = \frac{8}{x} \), these characteristics hold:

• If \( x \) is positive, \( y \) is uniquely positive, and vice-versa for negative values.
• No overlapping or repeated \( y \) values occur across different \( x \) inputs.

The one-to-one nature of \( y = \frac{8}{x} \) confirms it can undergo an inverse operation, vital for solving equations and transforming functions. Understanding these properties helps in the comprehensive analysis of function behavior.