How To See If A Function Is One To One

A function is one-to-one, also known as injective, if each element of the range corresponds to exactly one element of the domain. In simpler terms, a function f is one-to-one if for every y in the range of f, there is only one x in the domain such that f(x) = y. Determining whether a function is one-to-one is a fundamental concept in mathematics, particularly in areas like calculus, algebra, and analysis.

Understanding One-to-One Functions

Before diving into the methods to check if a function is one-to-one, it's crucial to understand the basic definition and implications. A function f(x) is one-to-one if it satisfies the condition:

If f(a) = f(b), then a = b for all a and b in the domain of f.

This definition means that if two different inputs, a and b, produce the same output, then a and b must actually be the same value. Conversely, if a and b are different, then f(a) and f(b) must also be different.

Why One-to-One Functions Matter

One-to-one functions are essential for several reasons:

• Invertibility: A function has an inverse if and only if it is one-to-one. The inverse function reverses the mapping, taking the output back to its unique input.
• Uniqueness: In many applications, ensuring that each input maps to a unique output and vice versa is critical. This is particularly important in cryptography, coding theory, and data analysis.
• Mathematical Properties: One-to-one functions have special properties in calculus and analysis, such as in the context of monotonic functions and their derivatives.

Methods to Determine if a Function is One-to-One

There are several methods to determine whether a function is one-to-one. These methods include algebraic verification, graphical tests, and calculus-based approaches. Each method has its strengths and is suitable for different types of functions.

1. Algebraic Verification

The most direct way to prove that a function f(x) is one-to-one is by using the definition. This involves the following steps:

• Assume f(a) = f(b).
• Show that this assumption leads to a = b.

If you can algebraically manipulate the equation f(a) = f(b) to show that a must equal b, then the function is one-to-one.

Example 1: Linear Function

Consider the function f(x) = 3x + 5. To check if it's one-to-one:

• Assume f(a) = f(b).
• Then 3a + 5 = 3b + 5.
• Subtract 5 from both sides: 3a = 3b.
• Divide by 3: a = b.

Since f(a) = f(b) implies a = b, the function f(x) = 3x + 5 is one-to-one.

Example 2: Rational Function

Consider the function f(x) = (2x - 1) / (x + 3). To check if it's one-to-one:

• Assume f(a) = f(b).
• Then (2a - 1) / (a + 3) = (2b - 1) / (b + 3).
• Cross-multiply: (2a - 1)(b + 3) = (2b - 1)(a + 3).
• Expand both sides: 2ab + 6a - b - 3 = 2ab + 6b - a - 3.
• Simplify: 6a - b = 6b - a.
• Rearrange: 7a = 7b.
• Divide by 7: a = b.

Since f(a) = f(b) implies a = b, the function f(x) = (2x - 1) / (x + 3) is one-to-one.

Example 3: Non-One-to-One Function

Consider the function f(x) = x². To check if it's one-to-one:

• Assume f(a) = f(b).
• Then a² = b².
• Take the square root of both sides: a = ±b.

Here, a can be equal to b or -b. For example, f(2) = 4 and f(-2) = 4, but 2 ≠ -2. Therefore, the function f(x) = x² is not one-to-one.

2. Horizontal Line Test

The horizontal line test is a graphical method to determine if a function is one-to-one. The principle is simple:

• If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one.
• If no horizontal line intersects the graph at more than one point, the function is one-to-one.

How to Apply the Horizontal Line Test:

1. Graph the function: Accurately plot the graph of the function on a coordinate plane.
2. Draw horizontal lines: Imagine or draw horizontal lines across the graph.
3. Check for intersections: Observe how many times each horizontal line intersects the graph.

Example 1: Linear Function

Consider the function f(x) = 2x - 1. Graphing this function shows a straight line. No horizontal line will intersect this line more than once. Therefore, f(x) = 2x - 1 is one-to-one.

Example 2: Quadratic Function

Consider the function f(x) = x². The graph of this function is a parabola. Any horizontal line above the x-axis will intersect the parabola at two points. Therefore, f(x) = x² is not one-to-one.

Example 3: Cubic Function

Consider the function f(x) = x³. The graph of this function is a curve that increases monotonically. No horizontal line will intersect this curve more than once. Therefore, f(x) = x³ is one-to-one.

3. Calculus-Based Approach: Using Derivatives

Calculus provides a powerful tool to determine if a function is one-to-one, especially for differentiable functions. The idea is based on the function's monotonicity.

• If f'(x) > 0 for all x in the domain, then f(x) is strictly increasing and one-to-one.
• If f'(x) < 0 for all x in the domain, then f(x) is strictly decreasing and one-to-one.
• If f'(x) changes sign or equals zero in the domain, f(x) is not necessarily one-to-one.

Steps to Apply the Derivative Test:

1. Find the derivative: Calculate the first derivative f'(x) of the function.
2. Analyze the sign of the derivative: Determine the intervals where f'(x) is positive, negative, or zero.
3. Conclude based on monotonicity: If f'(x) is consistently positive or negative across the domain, the function is one-to-one.

Example 1: Exponential Function

Consider the function f(x) = eˣ.

• Find the derivative: f'(x) = eˣ.
• Analyze the sign: eˣ is always positive for all x.

Since f'(x) > 0 for all x, the function f(x) = eˣ is strictly increasing and one-to-one.

Example 2: Cubic Function

Consider the function f(x) = x³.

• Find the derivative: f'(x) = 3x².
• Analyze the sign: 3x² ≥ 0 for all x. However, f'(x) = 0 at x = 0.

Since f'(x) is non-negative and equals zero only at a single point, the function is still one-to-one. f(x) = x³ is strictly increasing.

Example 3: Function with Local Extrema

Consider the function f(x) = x³ - 3x.

• Find the derivative: f'(x) = 3x² - 3.
• Analyze the sign: f'(x) = 3(x² - 1) = 3(x - 1)(x + 1).

f'(x) is positive for x < -1 and x > 1, and negative for -1 < x < 1. This means the function is increasing, then decreasing, then increasing again. Therefore, f(x) = x³ - 3x is not one-to-one.

4. Composition of Functions

Another method, though less direct, involves the composition of functions. If you can find a function g(x) such that g(f(x)) = x for all x in the domain of f, then f(x) is one-to-one. In this case, g(x) is the inverse function of f(x).

Example:

Consider the function f(x) = 2x + 3. Let's find its inverse:

1. Replace f(x) with y: y = 2x + 3.
2. Swap x and y: x = 2y + 3.
3. Solve for y: y = (x - 3) / 2.

So, g(x) = (x - 3) / 2 is the inverse function. Now, let's verify:

• g(f(x)) = g(2x + 3) = ((2x + 3) - 3) / 2 = (2x) / 2 = x.

Since g(f(x)) = x, the function f(x) = 2x + 3 is one-to-one.

5. Examples and Counterexamples

Let's examine some functions to illustrate these methods further.

• Example 1: f(x) = sin(x) on the interval [0, 2π]

  • Algebraic Verification: Consider f(0) = sin(0) = 0 and f(π) = sin(π) = 0. Since 0 ≠ π, f(x) is not one-to-one.
  • Horizontal Line Test: The graph of sin(x) oscillates between -1 and 1. A horizontal line like y = 0.5 intersects the graph at multiple points.
  • Derivative Test: f'(x) = cos(x). cos(x) changes sign in the interval [0, 2π], so f(x) is not monotonic and not one-to-one.

• Example 2: f(x) = x³ + 5

  • Algebraic Verification: If f(a) = f(b), then a³ + 5 = b³ + 5. This simplifies to a³ = b³, which implies a = b.
  • Horizontal Line Test: The graph of x³ + 5 is a cubic function shifted vertically. No horizontal line intersects it more than once.
  • Derivative Test: f'(x) = 3x². Although f'(0) = 0, f'(x) ≥ 0 for all x, and the function is strictly increasing.

• Example 3: f(x) = |x|

  • Algebraic Verification: f(1) = |1| = 1 and f(-1) = |-1| = 1. Since 1 ≠ -1, f(x) is not one-to-one.
  • Horizontal Line Test: The graph of |x| is a V-shape. Any horizontal line above the x-axis intersects the graph at two points.
  • Derivative Test: The derivative is not defined at x = 0. For x < 0, f'(x) = -1, and for x > 0, f'(x) = 1. The derivative changes sign, indicating it is not one-to-one.

Practical Considerations

When determining whether a function is one-to-one, consider the following practical points:

• Domain Restriction: A function that is not one-to-one over its entire domain might be one-to-one if the domain is restricted. For example, f(x) = x² is not one-to-one over the entire real line, but it is one-to-one if we restrict the domain to x ≥ 0.
• Piecewise Functions: For piecewise functions, each piece must be one-to-one, and the ranges of the pieces must not overlap.
• Complexity: Some functions are too complex for algebraic verification. In such cases, graphical or calculus-based methods may be more practical.

Conclusion

Determining whether a function is one-to-one is a fundamental skill in mathematics with applications in various fields. By using algebraic verification, the horizontal line test, and calculus-based approaches, you can effectively analyze different types of functions. Understanding the conditions under which a function is one-to-one is essential for invertibility, uniqueness, and further mathematical analysis. Remember to consider the domain and complexity of the function when choosing the appropriate method.