One to One Functions | Definition & Tests

A one-to-one function, also known as an injective function, is a type of mathematical relationship where each unique input from the domain maps to a unique output in the co-domain. Think of it like assigning a unique identifier to every item: if two items have the same identifier, they must be the exact same item. 🎯

More formally, a function f: A → B is considered one-to-one if for any two elements x₁ and x₂ in the domain A:

• If f(x₁) = f(x₂), then it must imply x₁ = x₂.
• Alternatively, if x₁ ≠ x₂, then it must imply f(x₁) ≠ f(x₂).

How to Test if a Function is One-to-One 🧐

There are two primary methods to test if a function is one-to-one:

1. The Horizontal Line Test (HLT) 📏 This is a graphical test.

   • How it works: If you can draw any horizontal line that intersects the graph of the function at more than one point, then the function is not one-to-one. ❌
   • Conversely, if every horizontal line you draw intersects the graph at at most one point (meaning it intersects once or not at all), then the function is one-to-one. ✅

2. The Algebraic Test This method uses the formal definition.

   • How it works:
     1. Start by assuming f(x₁) = f(x₂).
     2. Use algebraic manipulation to show that this assumption forces x₁ to be equal to x₂.
     3. If you can consistently arrive at x₁ = x₂, the function is one-to-one. If you find cases where x₁ ≠ x₂ but f(x₁) = f(x₂) (like x₁ = 2 and x₂ = -2 for f(x) = x²), then it’s not one-to-one.

Examples to Understand One-to-One Functions 📖

Let’s look at some examples:

• The Absolute Value Function: f(x) = |x|

  • Test: Using the Horizontal Line Test.
    • If you plot f(x) = |x|, you’ll see a V-shaped graph.
    • Draw a horizontal line (for example, at y = 2). This line will intersect the graph at two points: x = 2 and x = -2.
    • Since different inputs (2 and -2) produce the same output (2), and a horizontal line intersects the graph at more than one point, f(x) = |x| fails the HLT.
  • Conclusion: f(x) = |x| is not a one-to-one function 🚫.

• Exponential Functions: f(x) = a^x (where a > 0 and a ≠ 1)

  • Test: The sources state that exponential functions are either continuously increasing (when a > 1, e.g., 2^x) or continuously decreasing (when 0 < a < 1, e.g., (1/2)^x) across their domain.
  • Rule: A function that is strictly increasing or strictly decreasing throughout its domain is always one-to-one.
  • Conclusion: Exponential functions like f(x) = 5^x or f(x) = (1/2)^x are one-to-one functions ✅. They will always pass the Horizontal Line Test.

• Logarithmic Functions: f(x) = log_a x (where a > 0 and a ≠ 1)

  • Test: Logarithmic functions are defined as the inverse of exponential functions. The sources establish that if a function is one-to-one, its inverse exists. Conversely, if a function’s inverse exists, the original function must be one-to-one.
  • Rule: Similar to exponential functions, logarithmic functions are also either continuously increasing (when a > 1) or continuously decreasing (when 0 < a < 1).
  • Conclusion: Logarithmic functions are one-to-one functions ✅.

Practice Questions with Solutions 📝

Question 1: Is the function f(x) = x² a one-to-one function on its entire domain R? Explain using the concept of one-to-one functions. 🧐

Solution: No, f(x) = x² is not a one-to-one function on its entire domain R.

• Reasoning (Algebraic): Let’s choose two different input values, for example, x₁ = 3 and x₂ = -3.
  • f(x₁) = f(3) = 3² = 9
  • f(x₂) = f(-3) = (-3)² = 9
  • Here, we have x₁ ≠ x₂ (since 3 ≠ -3), but f(x₁) = f(x₂) (both equal 9). This contradicts the definition of a one-to-one function, which requires different inputs to produce different outputs.
• Reasoning (Graphical - Horizontal Line Test): If you were to sketch the graph of f(x) = x² (a parabola opening upwards), you would observe that any horizontal line drawn above the x-axis (e.g., y = 4 or y = 9) intersects the parabola at two distinct points. For instance, the line y = 9 intersects at x = 3 and x = -3. Since a horizontal line can intersect the graph at more than one point, f(x) = x² fails the Horizontal Line Test and is therefore not one-to-one. 📏❌

Question 2: Consider the function f(x) = 7^x. Is this function one-to-one? Justify your answer. 🤔

Solution: Yes, the function f(x) = 7^x is a one-to-one function. ✅

• Reasoning:
  • The function f(x) = 7^x is an exponential function with a base a = 7. Since a > 1, the sources confirm that this type of function is an increasing function across its entire domain.
  • A fundamental property stated in the sources is that if a function is increasing (or decreasing) throughout its domain, then it is necessarily one-to-one.
  • Therefore, because f(x) = 7^x is a continuously increasing function, it guarantees that every unique input x will produce a unique output f(x). Graphically, this means no horizontal line will ever intersect its graph at more than one point, so it passes the Horizontal Line Test.