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Chapter 0: Problem 51

Find the domain of each function given below. $$g(x)=|x|+1$$

Short Answer

Expert verified
The domain of \( g(x) = |x| + 1 \) is \( (-fty, fty) \).

Step by step solution

01

- Understand the function

Identify and understand the given function. The function given is \( g(x) = |x| + 1 \). This is an absolute value function with a constant added.
02

- Recall the properties of absolute value

Recall that the absolute value function \( |x| \) is defined for all real numbers \( x \). There is no restriction on \( x \) for the absolute value function.
03

- Analyze the addition with constant

Understand that adding a constant (in this case, 1) does not affect the domain of the function \( |x| \). The domain remains the same as the original absolute value function.
04

- Write the domain

Since the absolute value function is defined for all real numbers and adding a constant does not change this, the domain of \( g(x) = |x| + 1 \) is all real numbers. Therefore, the domain is \( (-fty, fty) \).

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Key Concepts

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

Absolute Value Function
The absolute value function is written as \(|x|\). This function represents the distance of a number, \(x\), from zero on the number line. It's always non-negative.
For any real number, \(x\), the absolute value is defined as:
  • \(|x| = x\) if \(x \geq 0\)
  • \(|x| = -x\) if \(x < 0\)
In simpler terms, \(|x|\) removes the sign of \(x\), making it positive if it wasn't already.
For example:
  • \(|3| = 3\)
  • \(|-3| = 3\)
Understanding this basic function is important in analyzing and plotting graphs that include absolute values.
Properties of Absolute Value
The absolute value has a few important properties that are helpful when solving problems:
  • \( |x| \geq 0 \) (non-negativity): The absolute value of any real number is never negative.
  • \( |x| = 0 \) if and only if \( x = 0 \): This means the only number whose absolute value is zero is zero itself.
  • \( |xy| = |x| |y| \): The absolute value of a product is the product of the absolute values.
  • \( |x + y| \leq |x| + |y| \) (triangle inequality): The absolute value of a sum is less than or equal to the sum of the absolute values.
These properties help in simplifying and understanding expressions that involve absolute values.
Domain of a Function
The domain of a function is the set of all possible input values (\(x\)-values) that the function can accept.
For the function \(g(x) = |x| + 1\), you need to determine which values of \(x\) can be plugged into the function without making it undefined.
Since the absolute value function \(|x|\) is defined for all real numbers and adding 1 does not restrict any values:
  • The domain of \(g(x) = |x| + 1\) is all real numbers, expressed as \( (-\infty, \infty) \).
This means you can input any real number into this function, and it will output a real number.
Real Numbers
Real numbers include all the numbers on the number line. This set is composed of rational numbers (like 2, 4/5, or -0.3) and irrational numbers (like \(\sqrt{2}\), \(\pi\), or \(e\)). Some important subsets of real numbers include:
  • Integers: These are positive and negative whole numbers, including zero (e.g., -2, 0, 7).
  • Rational numbers: Numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
  • Irrational numbers: Numbers that cannot be written as simple fractions (e.g., \(\pi\), \(\sqrt{2}\)).
Understanding real numbers is key because the domain of many mathematical functions, including the absolute value function, typically involves real numbers.

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