Which Of The Following Graphs Represents A One-to-one Function

In the realm of mathematics, particularly in the study of functions, the concept of a one-to-one function, also known as an injective function, holds significant importance. A one-to-one function is a function where each element of the range is associated with exactly one element of the domain. In simpler terms, each x-value corresponds to a unique y-value, and vice versa. Determining which graph represents a one-to-one function is a fundamental skill in understanding function properties and their graphical representations.

Understanding Functions and Their Graphs

Before diving into the specifics of identifying one-to-one functions from their graphs, it's essential to establish a foundational understanding of functions and their graphical representations.

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The set of inputs is called the domain, and the set of permissible outputs is called the range.

The graph of a function is a visual representation of the function's behavior on a coordinate plane. Each point on the graph represents an ordered pair (x, y), where x is an element of the domain and y is the corresponding element of the range.

Defining One-to-One Functions

A one-to-one function, or an injective function, is a function where each element of the range is associated with exactly one element of the domain. Mathematically, a function f is one-to-one if and only if for any x₁ and x₂ in the domain of f, if f( x₁) = f( x₂), then x₁ = x₂.

In simpler terms, a function is one-to-one if no two different x-values produce the same y-value. This property can be visually verified using the horizontal line test on the graph of the function.

The Horizontal Line Test

The horizontal line test is a graphical method used to determine whether a function is one-to-one. It states that a function is one-to-one if and only if no horizontal line intersects the graph of the function more than once.

To apply the horizontal line test, draw several horizontal lines across the graph of the function. If any horizontal line intersects the graph at more than one point, then the function is not one-to-one. If no horizontal line intersects the graph at more than one point, then the function is one-to-one.

Identifying One-to-One Functions from Graphs: Examples

Let's illustrate the process of identifying one-to-one functions from their graphs with a few examples:

Example 1: Linear Function

Consider the linear function f( x) = 2x + 1. The graph of this function is a straight line with a slope of 2 and a y-intercept of 1.

Applying the horizontal line test, we can see that no horizontal line intersects the graph more than once. Therefore, the linear function f( x) = 2x + 1 is a one-to-one function.

Example 2: Quadratic Function

Consider the quadratic function f( x) = x². The graph of this function is a parabola with its vertex at the origin (0, 0).

Applying the horizontal line test, we can see that many horizontal lines intersect the graph at two points. For example, the horizontal line y = 4 intersects the graph at the points (-2, 4) and (2, 4). Therefore, the quadratic function f( x) = x² is not a one-to-one function.

Example 3: Cubic Function

Consider the cubic function f( x) = x³. The graph of this function is a curve that passes through the origin (0, 0) and extends to both positive and negative infinity.

Applying the horizontal line test, we can see that no horizontal line intersects the graph more than once. Therefore, the cubic function f( x) = x³ is a one-to-one function.

Example 4: Absolute Value Function

Consider the absolute value function f( x) = |x|. The graph of this function is a V-shaped curve with its vertex at the origin (0, 0).

Applying the horizontal line test, we can see that many horizontal lines intersect the graph at two points. For example, the horizontal line y = 2 intersects the graph at the points (-2, 2) and (2, 2). Therefore, the absolute value function f( x) = |x| is not a one-to-one function.

Example 5: Exponential Function

Consider the exponential function f( x) = eˣ. The graph of this function is a curve that increases rapidly as x increases and approaches 0 as x decreases.

Applying the horizontal line test, we can see that no horizontal line intersects the graph more than once. Therefore, the exponential function f( x) = eˣ is a one-to-one function.

Example 6: Function with a Restricted Domain

The domain of a function plays a critical role in determining whether a function is one-to-one. Consider the function f( x) = x² again. We have already established that over the entire real number line, this function is not one-to-one. However, if we restrict the domain to x ≥ 0, then the function becomes one-to-one.

The graph of f( x) = x² for x ≥ 0 is only the right half of the parabola. Applying the horizontal line test to this restricted graph, we see that no horizontal line intersects the graph more than once. Therefore, f( x) = x² is a one-to-one function when its domain is restricted to x ≥ 0.

Example 7: Piecewise Function

Consider the piecewise function defined as follows:

f( x) = x, if x < 0 x², if x ≥ 0

For x < 0, the function is a straight line with a slope of 1, and for x ≥ 0, the function is a parabola. The graph of this function combines a straight line and a parabolic segment.

Applying the horizontal line test, we can see that horizontal lines above the x-axis will intersect the parabolic segment twice. Therefore, this piecewise function is not one-to-one.

Example 8: A Trigonometric Function

Consider the sine function, f( x) = sin(x). The graph of the sine function is a wave that oscillates between -1 and 1.

Applying the horizontal line test, we see that any horizontal line between -1 and 1 will intersect the graph multiple times. Therefore, the sine function is not one-to-one over its entire domain. However, similar to the quadratic function, we can restrict the domain to make it one-to-one, such as restricting x to the interval [-π/2, π/2].

Example 9: Rational Function

Consider the rational function f( x) = 1/x. The graph of this function is a hyperbola.

Applying the horizontal line test, no horizontal line intersects the graph more than once. Therefore, the rational function f( x) = 1/x is a one-to-one function.

Example 10: Constant Function

Consider the constant function f( x) = c, where c is a constant. The graph of this function is a horizontal line at y = c.

Applying the horizontal line test, this horizontal line intersects itself infinitely many times. Therefore, a constant function is not one-to-one.

Why Does the Horizontal Line Test Work?

The horizontal line test is a direct visual application of the definition of a one-to-one function. If a horizontal line intersects the graph of a function at more than one point, it means that there are at least two different x-values that produce the same y-value. This violates the condition that each y-value corresponds to exactly one x-value, thus indicating that the function is not one-to-one.

Conversely, if no horizontal line intersects the graph more than once, it means that each y-value corresponds to a unique x-value, satisfying the condition for a one-to-one function.

Domain Restrictions and One-to-One Functions

As demonstrated in Example 6 and Example 8, it's crucial to remember that a function that is not one-to-one over its entire domain may become one-to-one when its domain is restricted. This is because restricting the domain can eliminate the portions of the graph that cause horizontal lines to intersect more than once.

How to Determine the Appropriate Domain Restriction:

• Identify Problem Areas: Look for sections of the graph that cause the function to fail the horizontal line test. These are typically regions where the function increases and then decreases (or vice versa).
• Choose a Section: Select a section of the graph where the function is either strictly increasing or strictly decreasing.
• Define the Domain: Determine the corresponding x-values for the chosen section. This defines the restricted domain.

For example, to make f( x) = x² one-to-one, we can restrict the domain to x ≥ 0 or x ≤ 0. For the sine function, restricting the domain to [-π/2, π/2] or [π/2, 3π/2] are common choices to make it one-to-one.

Applications of One-to-One Functions

One-to-one functions have important applications in various areas of mathematics and computer science:

• Inverse Functions: A function has an inverse if and only if it is one-to-one. The inverse function "undoes" the original function, mapping each element of the range back to its corresponding element in the domain. Inverse functions are critical in solving equations and understanding functional relationships.
• Cryptography: One-to-one functions are used in cryptography to encrypt and decrypt messages. The encryption process transforms the original message into an unreadable form using a one-to-one function, and the decryption process uses the inverse function to recover the original message.
• Database Management: One-to-one relationships are used in database management to ensure data integrity. A one-to-one relationship between two tables means that each record in one table is associated with exactly one record in the other table.

Common Mistakes to Avoid

• Confusing One-to-One with Onto: A one-to-one function is not the same as an onto (surjective) function. An onto function is a function where every element in the range is mapped to by at least one element in the domain. A function can be one-to-one, onto, both (bijective), or neither.
• Not Applying the Horizontal Line Test Correctly: Ensure that you draw enough horizontal lines to cover all possible y-values in the range of the function. A single successful horizontal line does not guarantee that the function is one-to-one.
• Ignoring Domain Restrictions: Always consider the domain of the function when determining whether it is one-to-one. A function may be one-to-one on a restricted domain but not on its entire natural domain.
• Assuming All Functions are One-to-One: Many functions are not one-to-one. It's crucial to apply the horizontal line test or other methods to verify whether a function is one-to-one.
• Thinking Vertical Line Test Guarantees One-to-One: The vertical line test determines if a relation is a function, not if the function is one-to-one.

Conclusion

Determining whether a graph represents a one-to-one function is a fundamental skill in understanding function properties. The horizontal line test provides a simple and effective graphical method for identifying one-to-one functions. By carefully applying the horizontal line test and considering domain restrictions, one can accurately determine whether a function is one-to-one. Remember that understanding one-to-one functions is essential for various applications in mathematics, computer science, and other fields. Understanding one-to-one functions allows for a deeper comprehension of inverse functions, which are crucial in many mathematical operations and real-world applications. By mastering this concept, you solidify your understanding of the core principles underlying functions and their relationships.