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Chapter 1: Problem 61

True or false? Every graphically specified function can also be specified numerically. Explain.

Short Answer

Expert verified
False. While most graphically specified functions can be specified numerically, there are certain discrete functions that may not be easily specified numerically through an algebraic expression or equation.

Step by step solution

01

Define Graphically Specified Functions

A graphically specified function is a function that is defined by a graph instead of an algebraic expression or equation. With the graph, we can visually identify the input (x values) and the corresponding output (y values) for any point on the graph.
02

Define Numerically Specified Functions

A numerically specified function is a function that has a fixed set of ordered pairs (x, y) derived from an equation or algebraic expression. These functions can often be displayed in a table, which lists the input values (x) and the corresponding output values (y).
03

Evaluate if all Graphically Specified Functions can be Numerically Specified

Since functions can be continuous or discrete, we will examine both cases: 1. Continuous Functions: A continuous function is a function that has no gaps or breaks in its graph. Most continuous functions can be represented by an equation or algebraic expression, which can then be used to generate a set of ordered pairs. In this case, a graphically specified function can be specified numerically. Examples: \(y = x^2\), \( y = \sin(x) \) 2. Discrete Functions: A discrete function is a function that is only defined at particular points. Some discrete functions can be generated by algebraic expressions, while others may not have an algebraic representation. Examples: A sequence of prime numbers, tax bracket data
04

Conclusion

In most cases, graphically specified functions can be specified numerically. However, there are certain discrete functions that may not be easily specified numerically through an algebraic expression or equation. Therefore, the statement is not always true, and the answer is False.

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