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Chapter 2: Problem 8

Fill in the blank. The function \(f(x)=b\) is a _______ function.

Short Answer

Expert verified
The function \(f(x) = b\) is a constant function.

Step by step solution

01

Analyze the Function

Consider the function carefully. In the given function, the value of the function is a constant: it does not depend on the input variable \(x\). This means no matter what value of \(x\) is used, the function will always yield the same value, \(b\).
02

Identify the Type of Function

Understand the definition of the function type. A function that returns the same value regardless of the input is known as a constant function. This is because its output remains constant (unchanging).
03

Fill in the Blank

Based on the analysis, the blank should be filled with the term 'constant'. Thus, the function \(f(x) = b\) is a constant function.

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Key Concepts

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

constant function
A constant function is a type of function where the output value remains the same regardless of the input. Simply put, no matter what value you plug in for the variable \(x\), the function will always return the same constant value, typically denoted as \(b\). For example, with \(f(x) = 5\), the output is always 5, no matter if \(x\) is 0, 1, 100, or any other number. This characteristic makes them quite straightforward and easy to understand, as their graph is always a horizontal line.
function analysis
Analyzing a function means understanding how it behaves. For constant functions, this is quite simple. Since the function does not change with \(x\), its graph on a coordinate plane is a flat, horizontal line. The height of this line corresponds to the constant value \(b\) of the function. For instance, if our function is \(f(x) = 3\), then the graph is a horizontal line crossing the y-axis at 3. There's no need to worry about the line ever moving up or down, no matter how \(x\) changes.
function definition
A mathematical function can be seen as a rule that assigns exactly one output to each input. For a constant function, this rule is simple: every output is the same, no matter the input. We denote constant functions generally as \(f(x) = b\), where \(b\) is a fixed number. This differs from other types of functions where the output can vary based on the input. Because of its simplicity, constant functions serve as a foundational concept when learning about more complex function types.
precalculus
Precalculus functions as a preparatory stage for studying calculus. In precalculus, students learn about various kinds of functions, including constant functions. Understanding constant functions is critical as they form the building blocks for understanding linear functions, quadratic functions, and other more complicated forms鈥攅ven in calculus. By mastering constant functions and their simple behavior, students can build a solid foundation, making it easier to grasp more complex topics in future mathematical studies.

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Most popular questions from this chapter

Use transformations to graph each function and state the domain and range. $$y=3|x|-200$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\left|x^{2}-3\right|$$

Express \(\frac{-2+i}{4+2 i}\) in the form \(a+b i\)

Select two numbers \(a\) and \(b\). Then define a piecewise function using different formulas on the intervals \((-\infty, a],(a, b),\) and \([b, \infty)\) so that the graph docs not "jump" at \(a\) or \(b\). Give your function to a classmate to graph and check.

Use transformations to graph each function and state the domain and range. $$y=-\sqrt{x-3}+1$$
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