One-to-One Functions Calculator

One-to-One Function Tests:

\[ \text{1. } f'(x) > 0 \text{ or } f'(x) < 0 \text{ for all x} \] \[ \text{2. Horizontal Line Test: No horizontal line intersects graph more than once} \]

g(x) Result:

h(x) Result:

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1. What is a One-to-One Function?

Definition: A function is one-to-one (injective) if each output corresponds to exactly one input. No two different inputs produce the same output.

Purpose: One-to-one functions are important in mathematics because they have inverse functions and preserve uniqueness in mappings.

2. How to Determine if a Function is One-to-One

There are two main methods:

\[ \text{1. Derivative Test: } f'(x) > 0 \text{ or } f'(x) < 0 \text{ for all x} \] \[ \text{2. Horizontal Line Test: No horizontal line intersects graph more than once} \]

Explanation:

If a function is always increasing or always decreasing (strictly monotonic), it's one-to-one
If any horizontal line crosses the graph more than once, the function is not one-to-one

3. Importance of One-to-One Functions

Details: One-to-one functions are essential for creating inverse functions, solving equations uniquely, and in many applications like cryptography and data compression.

4. Using the Calculator

Tips: Enter your functions g(x) and h(x) in standard mathematical notation. The calculator will analyze them for one-to-one properties.

5. Frequently Asked Questions (FAQ)

Q1: Are all linear functions one-to-one?
A: Yes, all non-constant linear functions (f(x) = mx + b, m ≠ 0) are one-to-one.

Q2: Is f(x) = x² one-to-one?
A: No, because both x and -x produce the same output (e.g., 2² = (-2)² = 4).

Q3: Can a function be one-to-one if it's not continuous?
A: Yes, one-to-oneness doesn't require continuity, though continuous functions must be strictly monotonic to be one-to-one.

Q4: How does the derivative test work?
A: If the derivative is always positive (always increasing) or always negative (always decreasing), the function is one-to-one.

Q5: What's an example of a one-to-one function?
A: f(x) = x³ is one-to-one because it's always increasing (f'(x) = 3x² ≥ 0 and only 0 at x=0 but still strictly increasing).