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Chapter 12: Problem 94

Determine the domain of each function. $$k(t)=|-t|$$

Short Answer

Expert verified
The domain of the function \(k(t) = |-t|\) is \((-\infty, +\infty)\).

Step by step solution

01

Determine the absolute value function

The given function is k(t) = | -t |. This is an absolute value function, which takes the form of |x|. The absolute value function is defined for any real number as the distance from a number to zero on a number line.
02

Compute the domain of the function

The absolute value function is defined for all real numbers because the distance to zero can always be calculated for any real number. Therefore, the domain of k(t) = | -t | is all real numbers.
03

Express the domain in interval notation

The domain of the function is all real numbers, which can be expressed in interval notation as k(t) = \((-\infty, +\infty)\). So, the domain of the function k(t) = | -t | is \((-\infty, +\infty)\).

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Key Concepts

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

Understanding Absolute Value Functions
An absolute value function, like \(k(t) = |-t|\), represents the non-negative distance of a number from zero on the number line. This function is identified by the \(|x|\) notation.

Here are some key characteristics of absolute value functions:
  • They always return positive outputs or zero, since distance cannot be negative.
  • The graph of an absolute value function forms a V-shape.
  • The function is defined for every real number, meaning there are no input restrictions.


For the specific function \(k(t) = |-t|\), no matter what real number \(t\) is, taking the absolute value ensures the output is the distance from zero. This ensures that \(k(t)\) is valid for any choice of \(t\).
Exploring Real Numbers
Real numbers are a fundamental concept in mathematics, encompassing a wide range of values:

  • They include all the numbers that can be found on the number line.
  • This set encompasses rational numbers (like fractions) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).
  • Real numbers can be positive, negative, or zero.


The function \(k(t) = |-t|\) is defined across all real numbers, indicating that any real number can be an input. This universality is one of the reasons why the domain can be expressed as all real numbers without restriction.
Using Interval Notation for Domain
Interval notation is a concise way to express the set of numbers that form the domain of a function.

For the function \(k(t) = |-t|\), the domain is all real numbers. In interval notation, this range is represented as \((-fty, +fty)\). This means:
  • It starts from negative infinity, indicating it includes all negative numbers.
  • It extends to positive infinity, capturing all positive numbers.


This compact form is helpful for quickly understanding the extent of any function's domain. By writing \((-fty, +fty)\), we indicate that there are no breaks or gaps in the values that \(t\) can take.

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Most popular questions from this chapter

Use the transformation techniques to graph each of the following functions. $$h(x)=-|x+3|-2$$

Let \(h(x)=3 x^{2}-8 x+2\) and \(k(x)=2 x-3 .\) Finc a) \(\quad(h \circ k)(x)\) b) \(\quad(k \circ h)(x)\) c) \(\quad(k \circ h)(0)\)

To consult with an attorney costs \(\$ 35\) for every \(10 \mathrm{min}\) or fraction of this time. Let \(\mathcal{C}(t)\) represent the cost of meeting an attorney, and let \(t\) represent the length of the meeting, in minutes. Graph \(C(t)\) for meeting with the attorney for up to (and including) 1 hr.

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