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Chapter 1: Problem 77

Find the domain and range of each of the following functions. $$g(s)=1+\sqrt{s+1}$$

Short Answer

Expert verified
The domain of the function \( g(s)=1+\sqrt{s+1} \) is \( s \geq -1 \), and the range is all real numbers greater than or equal to 1.

Step by step solution

01

Finding the Domain

For the function \( g(s)=1+\sqrt{s+1} \), first let's find the domain. The domain will be all 's' values that ensure the square root will give a real number. This is when the radicand, \( s+1 \), is equal to or greater than zero. Hence, to find the domain, we solve \( s+1 \geq 0 \), leading to \( s \geq -1 \). This is the domain.
02

Finding the Range

Next, for finding the range, consider that the square root function outputs all non-negative real numbers, and '1' is being added to this. Hence, \( g(s) \) produces all real numbers that are more than or equal to 1. This is the range.

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Key Concepts

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

Square Root Function
The square root function is an interesting part of mathematics that deals with finding a number that, when multiplied by itself, gives the original number. In the given function, \( g(s) = 1 + \sqrt{s + 1} \), we encounter a square root.
  • The square root always requires non-negative numbers as its input, meaning the expression inside the square root, known as the radicand, must be zero or positive for real numbers.
  • Here, \( s+1 \) is the radicand, so \( s+1 \geq 0 \) determines valid input values for 's'.
  • Solving \( s+1 \geq 0 \) gives \( s \geq -1 \), establishing the domain of the function.
Understanding how square roots behave is key to defining the domain and comprehending the behavior of functions involving them.
Real Numbers
Real numbers encompass all numbers that can be found on the number line. These include integers, fractions, and decimals, covering both positive and negative numbers, plus zero.
When discussing functions like \( g(s) = 1 + \sqrt{s + 1} \), whether we talk about domain or range, real numbers play a critical role.
  • The result of a square root is always a non-negative real number.
  • For example, \( \sqrt{0} = 0 \) and \( \sqrt{4} = 2 \), both are real numbers.
  • This informs us that starting from \( s = -1 \), the function will return real results.
Therefore, the function outputs for all conceivable 's' values are within the realm of real numbers, always producing valid and tangible results.
Inequalities
Inequalities are mathematical expressions used to define the relationship between two values, often allowing us to find valid input or output ranges. They use symbols like \( \geq \), \( \leq \), \( > \), and \( < \).
In the exercise concerning \( g(s) = 1 + \sqrt{s+1} \), inequalities help us establish both domain and range.
  • For the domain: \( s+1 \geq 0 \) implies \( s \geq -1 \). This inequality ensures the radicand gives a real number result.
  • For the range: The inequality \( g(s) \geq 1 \) comes from the understanding that a square root result is never negative, and we are adding 1.
  • Thus, inequalities give us the ability to precisely list out conditions where functions behave as needed.
Understanding how to manipulate and apply inequalities allows for precise identification of domains and ranges in functions involving square roots.

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Most popular questions from this chapter

This set of exercises will draw on the ideas presented in this section and your general math background. What happens when you try to find the intersection of \(y=x\) and \(y=x+2\) algebraically? Graph the two lines on the same set of axes. Do they appear to intersect? Why or why not? This is an example of how graphs can help you to see things that are not obvious from algebraic methods. Examples such as this will be discussed in greater detail in a later chapter on systems of linear equations.

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