/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 find the domain and range of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 18

find the domain and range of the function. Use interval notation to write your result. $$ f(x)=\sqrt{2 x-3} $$

Short Answer

Expert verified
The domain of the function \( f(x)=\sqrt{2x-3} \) is \([1.5, +\infty)\) and the range is \([0, +\infty)\).

Step by step solution

01

Find the Domain

First, set the expression inside the square root \( 2x-3 \) greater than or equal to zero to find the valid x-values. It can be solved like this: \(2x - 3 \geq 0 \)\n Adding 3 to both sides of the inequality we get:\n \(2x \geq 3 \)\n Then, divide each side of the inequality by 2:\n \(x \geq 1.5 \)\nSo, all x-values 1.5 and greater are valid. Therefore, the domain in interval notation is \([1.5, +\infty)\).
02

Find the Range

The square root function cannot output negative values, and the smallest number it can output is 0 when \( x=1.5 \). Thus, any x-value greater than or equal to 1.5 will yield a y-value greater than or equal to 0. Therefore, the range in interval notation is \([0, +\infty)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Interval Notation
Interval notation is a way of representing a set of numbers that fall within a certain range. It uses brackets and parentheses to show which numbers are included and which are not.
  • Brackets, \([ \text{ and } ]\), are used when the endpoint is included in the interval. This is known as a "closed interval."
  • Parentheses, \(( \text{ and } )\), are used when the endpoint is not included, indicating an "open interval."
For example, \([1.5, +\infty)\) represents all numbers from 1.5 to infinity, including 1.5 but not infinity.
When writing the domain or range of a function in interval notation, it provides a clear and concise way to present all possible values of \(x\) or \(y\). Understanding this will help communicate the boundaries within which the function operates.
Square Root Function
A square root function involves taking the square root of an expression. It's represented by \(f(x) = \sqrt{x}\). While simple, it has unique characteristics to consider when analyzing its domain and range.
  • The square root function is only defined for non-negative inputs. This means that the expression inside the square root must be zero or positive.
  • The basic shape of the graph of a square root function starts at the origin (for the base function \(\sqrt{x}\)), and it gradually increases without ever decreasing, forming a curved shape.
  • For \(f(x) = \sqrt{2x - 3}\), the inside expression \(2x - 3\) must be non-negative for the function to produce real numbers, therefore the domain is derived by ensuring \(2x - 3 \geq 0\).
Understanding this function's nature is crucial in determining proper values for its domain and range.
Function Analysis
Function analysis involves understanding various aspects of a function, such as its domain, range, and general behavior. This skill is vital for interpreting mathematical models and real-world applications.
  • Domain: The domain of a function includes all possible input values \(x\) that the function can accept. For a square root function like \(f(x)=\sqrt{2x-3}\), we determine the domain by finding all \(x\) for which the expression \(2x-3\) is non-negative.
  • Range: The range includes all possible output values \(y\) the function can produce. Since a square root function outputs non-negative values, \([0, +\infty)\) is the range for \(f(x)=\sqrt{2x-3}\).
  • Behavior: Analyzing how the function behaves as \(x\) changes is also important. For the square root function, as \(x\) increases, the \(y\) value gradually increases, reflecting a pattern of slow growth.
By developing a strong grasp on function analysis, students can effectively tackle problems involving complex functions and accurately predict their characteristics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the limit. \(\lim _{x \rightarrow 3} \sqrt{x+6}\)

Find the limit (if it exists). \(\lim _{x \rightarrow 2} \frac{2-x}{x^{2}-4}\)

use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one. If it is, find its inverse function. $$ g(t)=\frac{t+3}{1-t} $$

MAKE A DECISION: REVENUE For groups of 80 or more people, a charter bus company determines the rate \(r\) (in dollars per person) according to the formula \(r=15-0.05(n-80), \quad n \geq 80\) where \(n\) is the number of people. (a) Express the revenue \(R\) for the bus company as a function of \(n .\) (b) Complete the table. (c) Criticize the formula for the rate. Would you use this formula? Explain your reasoning.

Find the limit (if it exists). \(\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.