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Chapter 0: Problem 40

Evaluate each of the functions at the given value of \(x\). $$f(x)=|x|, \quad x=\pi$$

Short Answer

Expert verified
The value of the function at \(x = \pi\) is \(\pi\).

Step by step solution

01

Understand the Absolute Value Function

The given function is an absolute value function, defined as \(f(x) = |x|\). This means that the function takes any input \(x\) and returns its non-negative value.
02

Substitute the Given Value of \(x\)

Insert the given value \(x = \pi\) into the function. The function becomes \(f(\pi) = |\pi|\).
03

Evaluate the Absolute Value

\(\pi\) (pi) is approximately 3.14, which is already a non-negative number. Therefore, the absolute value of \(\pi\) is simply \(\pi\). So, \(f(\pi) = \pi\).

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Key Concepts

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

Absolute Value
An absolute value function is a mathematical function that returns the non-negative value of any number you input. In mathematical terms, the absolute value of a number \(x\) is written as \(|x|\). This function helps in understanding how far a number is from zero on the number line, without considering direction. The key rule to remember is:
  • If the number is positive or zero, the absolute value is the same as the number.
  • If the number is negative, the absolute value is the number without the negative sign.
For example, \(|-5| = 5\) and \(|3| = 3\). This is exactly how we approach the function \(f(x) = |x|\) in the exercise.
Function Evaluation
Function evaluation is the process of determining the output of a function for a specific input value. It involves substituting the input value into the function and then simplifying it.
In the exercise, we evaluate the function \(f(x) = |x|\) at \(x = \pi\). Here is how:
  • First, take the function \(f(x) = |x|\).
  • Next, substitute \(x\) with \(\pi\), which gives us \(f(\pi) = |\pi|\).
  • Since \(\pi\) is approximately 3.14 and already non-negative, \(|\pi| = \pi\).
This gives us the evaluated result of the function at \(x = \pi\): \(f(\pi) = \pi\).
蟺 (pi)
Pi (\(\pi\)) is a fundamental mathematical constant, approximately equal to 3.14159. It's the ratio of a circle's circumference to its diameter. Pi is an irrational number, meaning it cannot be exactly expressed as a simple fraction and its decimal representation never ends or repeats.
In practical terms, pi shows up in various fields of science and engineering, especially where circular and periodic phenomena are involved.
In the given exercise, \(\pi\) is used as the value to evaluate the function. We substitute \(\pi\) into the absolute value function, noting that the absolute value of a positive number like \(\pi\) remains unchanged, therefore simplifying our calculations.

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Most popular questions from this chapter

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