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Chapter 0: Problem 39

Evaluate each of the functions at the given value of \(x\). $$f(x)=|x|, \quad x=10^{-2}$$

Short Answer

Expert verified
The evaluated value is \(10^{-2}\).

Step by step solution

01

Understand the function

The given function is the absolute value function, represented as \(f(x) = |x|\). This function takes any real number and returns its non-negative value.
02

Substitute the value of \(x\)

Substitute the given value of \(x\) which is \(10^{-2}\) into the function: \[f(10^{-2}) = |10^{-2}|\]
03

Evaluate the absolute value

The absolute value of a positive number is the number itself. Therefore, \(|10^{-2}| = 10^{-2}\).

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Key Concepts

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

absolute value
The absolute value function is a fundamental mathematical concept. It is represented by vertical bars around a number, such as \(|x|\). The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value is always a non-negative number. For instance, \(|-5| = 5\) and \(|5| = 5\). So, the absolute value function converts negative numbers to positive numbers while keeping positive numbers unchanged.
function evaluation
Function evaluation involves substituting a given value into a function to find the result. For example, if we have the function \(f(x) = |x|\) and we are given \(x = 10^{-2}\), we substitute \(x\) with \(10^{-2}\). Therefore, we compute \(f(10^{-2}) = |10^{-2}|\). This process helps us understand how a function behaves for specific inputs. By evaluating the function at a given point, we can determine the output or y-value corresponding to that x-value.
positive number
A positive number is any number that is greater than zero. In the context of the absolute value function, when evaluating \(f(x) = |x|\) and x is a positive number, the absolute value function will simply return the number itself. For example, when evaluating \(f(10^{-2}) = |10^{-2}|\), since \(10^{-2}\) is positive, the result is \(10^{-2}\). Remember, the absolute value of x is always a positive number or zero, making it a useful tool for ensuring non-negative results in various mathematical computations.

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