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Chapter 2: Problem 6

Yes or No? If No, give a reason. Let \(f\) be a function. (a) Is it possible that \(f(1)=5\) and \(f(2)=5 ?\) (b) Is it possible that \(f(1)=5\) and \(f(1)=6 ?\)

Short Answer

Expert verified
(a) Yes; (b) No, a function can't have different values for the same input.

Step by step solution

01

Identification of Function Property

In part (a), we need to determine if a function can take the same value at different inputs, while in part (b), we need to check if a function can take different values at the same input. For all functions, each input has exactly one output, so it's essential to analyze what is being asked.
02

Analyzing Part (a)

Check if a function can have the same output for different inputs. Here, the function gives the same output value (5) for different inputs (1 and 2). This is possible because a function can map multiple inputs to the same output.
03

Analyzing Part (b)

Check if a function can have different outputs for the same input. Here, two different output values (5 and 6) are given for the same input (1). This is not possible because a function must have a single, unique output for every input.

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Key Concepts

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

Unique Outputs
In mathematics, a function is a special kind of relation that assigns exactly one output to each input. This characteristic is crucial when understanding why
  • for part (a), the function can have the same output for different inputs
  • and for part (b), the same input cannot have different outputs.

Think of a function like a vending machine. You select a button (input), and the vending machine gives you a specific snack (output). Even if you push multiple buttons (different inputs) and they all dispense the same snack (same output), it's functioning correctly. However, if you push the same button twice and get different snacks each time, something is wrong; the machine isn't functioning as intended. The same principle applies to functions, as each input must always lead to the same single output.
Mapping Inputs
Mapping in the context of functions refers to how each input is assigned an output. A key property of functions is that this mapping must be consistent: a specific input always leads to the same output.
The concept of mapping helps us understand the conditions presented:
  • In part (a), mapping different inputs (1 and 2) to the same output (5) is acceptable. This is because mapping doesn't require different inputs to have different outputs; multiple inputs can map to a single common output.
  • In part (b), the attempt to map the same input (1) to different outputs (5 and 6) contradicted the basic functioning property.

Consider mapping as drawing a line from each input circle to an output circle. As long as no input circle has two lines coming from it indicating different outputs, the mapping remains valid as a function.
Function Definition
The definition of a function is what sets it apart and governs its behavior in mathematics. According to this definition, a function assigns one and only one output to each input in its domain.
This core principle helps us resolve questions like those in the exercise:
  • In part (a), since different numbers (1 and 2) map to the same number (5), there isn鈥檛 any conflict with the function definition.
  • Part (b) violates the definition because the same input (1) cannot map to two different outputs (5 and 6), breaking the fundamental rule of having a single output for every input.

Understanding this definition is crucial because it helps us determine what is and isn't possible within the framework of functions. That's why functions are such a vital concept in mathematics, underpinning everything from simple algebra to complex calculus.

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Most popular questions from this chapter

A home owner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period beginning on a Sunday.

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(x)=c x^{2}\) (a) \(c=1, \frac{1}{2}, 2,4 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=1,-1,-\frac{1}{2},-2 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

An appliance dealer advertises a \(10 \%\) discount on all his washing machines. In addition, the manufacturer offers a \(\$ 100\) rebate on the purchase of a washing machine. Let \(x\) represent the sticker price of the washing machine. (a) Suppose only the \(10 \%\) discount applies. Find a function \(f\) that models the purchase price of the washer as a function of the sticker price \(x\). (b) Suppose only the \(\$ 100\) rebate applies. Find a function \(g\) that models the purchase price of the washer as a function of the sticker price \(x\). (c) Find \(f \circ g\) and \(g \circ f .\) What do these functions represent? Which is the better deal?

For every integer \(n\), the graph of the equation \(y=x^{n}\) is the graph of a function, namely \(f(x)=x^{n} .\) Explain why the graph of \(x=y^{2}\) is not the graph of a function of \(x .\) Is the graph of \(x=y^{3}\) the graph of a function of \(x ?\) If so, of what function of \(x\) is it the graph? Determine for what integers \(n\) the graph of \(x=y^{n}\) is a graph of a function of \(x\)

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(x)=\frac{1}{x^{n}}\) (a) \(n=1,3 ;[-3,3]\) by \([-3,3]\) (b) \(n=2,4 ;[-3,3]\) by \([-3,3]\) (c) How does the value of \(n\) affect the graph?
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