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Chapter 1: Problem 312

True or False? Justify your answer with a proof or a counterexample. A relation that passes the horizontal and vertical line tests is a one-to-one function.

Short Answer

Expert verified
True, because passing both tests satisfies the criteria for a one-to-one function.

Step by step solution

01

Understanding the Vertical Line Test

The vertical line test is a method used to determine if a relation is a function. A relation passes the vertical line test if and only if no vertical line intersects the graph of the relation at more than one point. Thus, a relation passing this test is a function because for each input (x-value), there is only one output (y-value).
02

Understanding the Horizontal Line Test

The horizontal line test is a method used to determine if a function is one-to-one. A function passes the horizontal line test if and only if no horizontal line intersects the graph of the function at more than one point. This means that each output (y-value) is produced by only one input (x-value), qualifying the function as one-to-one.
03

Combining Both Tests

For a graph to pass both the vertical and horizontal line tests, it must be a function (by the vertical line test) and a one-to-one function (by the horizontal line test). Consequently, if a relation passes both tests, it confirms that it is indeed a one-to-one function because it satisfies the conditions for being a function and having unique mappings from inputs to outputs.
04

Conclusion

Given that a relation passing both the vertical and horizontal line tests satisfies the criteria for being a one-to-one function, the statement is True. Passing the vertical line test confirms it is a function, and passing the horizontal line test proves it is one-to-one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Line Test
The vertical line test is a straightforward method to determine if a relation is actually a function. Imagine drawing vertical lines across the graph of a given relation. If at any point, a vertical line touches the graph at more than one spot, then the relation is not a function. This behavior indicates that for a single input, there might be multiple outputs, which violates the definition of a function.
  • A successful vertical line test means the graph is a function: every x-value has only one y-value.
  • Failing this test implies the presence of multiple y-values for a single x-value.
Understanding this concept is key. It forms the basis for understanding more advanced concepts about functions. It ensures that the function behaves predictably with each input value mapping to a single output.
Horizontal Line Test
Similar to the vertical line test, the horizontal line test helps determine if a function is one-to-one (injective). When we apply this test, we draw horizontal lines across the graph. If any horizontal line crosses the graph more than once, the function is not one-to-one, meaning the same output value corresponds to multiple input values.
  • Passing the horizontal line test guarantees that each y-value comes from a unique x-value.
  • If the test is failed, a function is not one-to-one despite possibly still being a function.
Why is this important? In mathematics, one-to-one functions are crucial because they have unique inverses. Understanding whether a function passes this test guides us in exploring these inverses and deeper relationships.
Function
In its simplest form, a function is a relation where each input has a single output. This concept is fundamental in mathematics and provides the groundwork for analyzing almost any mathematical operation.
  • A function must assign exactly one output for each input.
  • The definition includes relations that pass the vertical line test.
From basic algebra to complex analysis, functions appear everywhere, helping us model real-world situations and problems. Grasping the core idea of a function is essential as it links inputs to outputs in a consistent and predictable manner.

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Most popular questions from this chapter

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse \(f^{-1}(x)\) of the function. Justify your answer. $$ f(x)=\frac{1}{x} $$

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