select all of the following graphs which are one-to-one functions.

Guri Bee

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20-10-2024

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Unveiling One-to-One Functions: A Visual Exploration

In the world of mathematics, functions are like machines that take an input and produce an output. A special type of function, known as a one-to-one function, has a unique characteristic: each input maps to a distinct output, and vice versa. This means no two inputs can share the same output.

To help visualize this concept, let's explore some common graphs and determine if they represent one-to-one functions. We'll draw upon insights from a GitHub discussion where users grapple with this very question.

Question: From the following graphs, select all that are one-to-one functions.

(a) A straight line with a positive slope. (b) A parabola that opens upwards. (c) A circle. (d) A graph that is always increasing.

Answer: (a) and (d) are one-to-one functions.

Explanation:

• (a) A straight line with a positive slope: This graph represents a linear function, where the output changes at a constant rate with respect to the input. Since each input maps to a unique output, this is a one-to-one function.

• (b) A parabola that opens upwards: This graph represents a quadratic function. While the output changes at a non-constant rate, the issue arises when considering the symmetry of the parabola. Notice that for different inputs, the parabola outputs the same value, violating the one-to-one rule. For example, a parabola might have two distinct input values that correspond to the same output value at the vertex.

• (c) A circle: This graph represents a circular function. This fails the one-to-one test because for any given y-value, there are two distinct x-values that correspond to it. This is evident due to the symmetrical nature of the circle.

• (d) A graph that is always increasing: If a graph is always increasing, it means that as the input value increases, the output value always increases as well. This ensures that no two inputs can share the same output, making it a one-to-one function.

Real-World Analogy:

Imagine a vending machine. If it dispenses only one specific item for each button press (input), it represents a one-to-one function. However, if two buttons dispense the same item, then it's not a one-to-one function.

The Horizontal Line Test:

A handy tool for determining one-to-one functions visually is the Horizontal Line Test. If any horizontal line intersects the graph at more than one point, then the function is not one-to-one.

Example:

• A straight line with a positive slope: No horizontal line intersects the graph at more than one point.
• A parabola that opens upwards: A horizontal line can intersect the parabola at two points.

In Conclusion:

Understanding one-to-one functions is crucial in various fields, from mathematics and computer science to engineering and economics. The ability to recognize and analyze these functions is essential for solving complex problems and interpreting data effectively.

Note: This analysis is based on the information shared in the GitHub discussion, which was a valuable resource in helping us understand the concepts. Remember that the context of the discussion might provide additional information relevant to the problem.