One to One Functions | Definition & Tests

Absolutely! Let’s dive into one-to-one functions, making it easy to understand with definitions, tests, examples, and practice questions. 🎯

What is a One-to-One Function? 🤔

A one-to-one function, also known as an injective function, is a special type of mathematical relationship where each unique input from the domain maps to a unique output in the co-domain. Think of it like this:

• Imagine a class where every student has a unique student ID. No two different students can have the same ID. That’s a one-to-one relationship! ✅🧑‍🎓🆔
• If, however, two different students could have the same ID (e.g., student 1 and student 2 both have ID 123), then it’s not one-to-one. 🙅‍♀️

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. (Meaning, if the outputs are the same, the inputs had to be the same).
• Alternatively, if x₁ ≠ x₂, then it must imply f(x₁) ≠ f(x₂). (Meaning, different inputs always lead to different outputs).

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

There are two primary methods to check 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. ✅

   Example: The graph of f(x) = |x| is a V-shape. If you draw a horizontal line at y=2, it intersects the graph at x=2 and x=-2. Since it intersects at more than one point, f(x) = |x| is not 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 for f(x) = x², where f(2)=4 and f(-2)=4), then it’s not one-to-one.

Key Theorems about One-to-One Functions 📚

• Theorem on Monotonicity: If a function is strictly increasing or strictly decreasing throughout its domain, then the function is one-to-one.

  • Explanation: If a function is always going up (increasing) or always going down (decreasing), it will never return to the same y-value for different x-values. Think of a staircase: if you only ever go up, you won’t land on the same step twice unless you haven’t moved. ⬆️⬇️
  • Example: Exponential functions like f(x) = a^x (where a > 1) are strictly increasing, and f(x) = a^x (where 0 < a < 1) are strictly decreasing. Therefore, exponential functions are one-to-one functions. Similarly, logarithmic functions are also one-to-one. Linear functions like f(x) = ax + b (where a ≠ 0) are also either strictly increasing or strictly decreasing, making them one-to-one.

• Theorem on Inverse Functions: A function f is one-to-one if and only if its inverse function f⁻¹ exists.

  • Explanation: For an inverse function to exist, it must be possible to “reverse” the mapping uniquely. If two different inputs x₁ and x₂ map to the same output y, then when you try to go back from y, you wouldn’t know whether to go to x₁ or x₂. A one-to-one function guarantees this unique reverse mapping. ↩️

Examples to Enhance Understanding 💡

Let’s look at more examples and apply our tests:

• Linear Function: f(x) = 2x + 3

  • Algebraic Test: Assume f(x₁) = f(x₂).
    • 2x₁ + 3 = 2x₂ + 3
    • 2x₁ = 2x₂
    • x₁ = x₂
  • Conclusion: Since f(x₁) = f(x₂) implies x₁ = x₂, f(x) = 2x + 3 is a one-to-one function ✅. (It’s also strictly increasing, confirming this by the monotonicity theorem).

• Quadratic Function: f(x) = x²

  • Algebraic Test: Assume f(x₁) = f(x₂).
    • x₁² = x₂²
    • This means x₁ = x₂ or x₁ = -x₂.
    • Since x₁ does not have to be equal to x₂ (e.g., if x₁ = 2, then x₂ could be 2 or -2, and f(2) = f(-2) = 4), the condition x₁ = x₂ is not forced.
  • Horizontal Line Test: The graph of f(x) = x² is a parabola opening upwards. A horizontal line drawn above the x-axis will intersect the parabola at two points.
  • Conclusion: f(x) = x² is not a one-to-one function 🚫.

• Cubic Function: f(x) = x³

  • Algebraic Test: Assume f(x₁) = f(x₂).
    • x₁³ = x₂³
    • Taking the cube root of both sides, x₁ = x₂.
  • Horizontal Line Test: The graph of f(x) = x³ always passes the horizontal line test (each horizontal line intersects at most once).
  • Conclusion: f(x) = x³ is a one-to-one function ✅. (It’s also strictly increasing, as seen in the source).

Practice Questions with Solutions 📝

Question 1: Is the function f(x) = x + 5 a one-to-one function? Justify your answer using the algebraic test. 🤔

Solution: Yes, f(x) = x + 5 is a one-to-one function. ✅

• Justification (Algebraic Test):
  1. Assume f(x₁) = f(x₂).
  2. Substitute the function definition: x₁ + 5 = x₂ + 5.
  3. Subtract 5 from both sides: x₁ = x₂.
  4. Since the assumption f(x₁) = f(x₂) directly implies x₁ = x₂, the function satisfies the definition of a one-to-one function.

Question 2: Consider the function f(x) = (1/2)^x. Is this function one-to-one? Justify your answer using a relevant theorem. 🧐

Solution: Yes, the function f(x) = (1/2)^x is a one-to-one function. ✅

• Justification (Monotonicity Theorem):
  1. The function f(x) = (1/2)^x is an exponential function with a base a = 1/2.
  2. The sources state that when the base a is between 0 and 1 (i.e., 0 < a < 1), an exponential function is a decreasing function. Specifically, if x₁ < x₂, then a^x₁ > a^x₂.
  3. A key theorem states that if a function is an increasing or decreasing function, then it is one-to-one.
  4. Therefore, since f(x) = (1/2)^x is a strictly decreasing function, it is one-to-one. Graphically, it would pass the Horizontal Line Test.

Question 3: Is the relation represented by the graph of a circle, say x² + y² = r², a one-to-one function? Explain why or why not, referring to one of the graphical tests. ⭕

Solution: No, the graph of a circle x² + y² = r² does not represent a one-to-one function, because a circle is not even a function to begin with. 🚫

• Justification (Vertical Line Test):
  1. A preliminary test for any relation to be a function is the Vertical Line Test (VLT). If a vertical line intersects the graph at more than one point, the relation is not a function.
  2. If you draw a vertical line through a circle (except at its extreme left and right points), it will intersect the circle at two distinct points. This means for a single x input, there are two y outputs.
  3. Since the circle fails the Vertical Line Test, it is not a function. Consequently, it cannot be a one-to-one function.

This concludes the explanation of one-to-one functions! I hope the emojis and clear steps made it easier to grasp! 😊