/*! 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 30 Determine whether the equation r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

Determine whether the equation represents \(y\) as a function of \(x\). $$ y=\sqrt{x+5} $$

Short Answer

Expert verified
Yes, the equation \(y=\sqrt{x+5}\) does represent `y` as a function of `x`.

Step by step solution

01

Locate the Radicand

In the given equation \(y=\sqrt{x+5}\), the part that is a radicand is 'x + 5'. This means that the function is defined as long as \(x + 5 \geq 0\).
02

Find the Domain

In this step, the objective is to define the range of `x` values that can be plugged into the equation to receive a real number. Solve \(x + 5 \geq 0\) for `x`. This will give \(x \geq -5\). So, for any `x` greater than or equal to -5, the equation will hold a real value of `y`.
03

Validate the Function

To confirm that this equation represents a function, for each input `x` in the domain there should be exactly one output `y`. Given the domain is \(x \geq -5\) and having square root function, each input `x` will produce one and only one output of `y`. Thus, the given equation can be classified as a function.

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.

Radicand
In mathematics, the radicand is the term inside the square root sign. It's the value you are trying to find the square root of. In the equation \( y = \sqrt{x+5} \), the radicand is \( x+5 \). Understanding the radicand is crucial in finding out which values inside the square root can yield real numbers.
  • Radicands are important because you need to know if they're positive to make sure the square root results in a real number.
  • Real numbers are those that you can plot on the number line, unlike complex numbers which involve imaginary components.
For \( y = \sqrt{x+5} \), the radicand \( x+5 \) must be zero or positive, \( x+5 \geq 0 \), so that the square root can return a real number.
Domain
The domain of a function refers to all the possible input values (usually \( x \)) that can be used. It's essential to determine the domain to understand what values are allowed in the function to yield a real output.For the function \( y = \sqrt{x+5} \), determining the domain involves solving the inequality \( x + 5 \geq 0 \):
  • Subtract 5 from both sides to get \( x \geq -5 \).
  • This tells us that \( x \) can be any value starting from \(-5\) to infinity.
Therefore, the domain of this function is \( x \geq -5 \). Any \( x \)-value within these numbers will allow the function to work properly and provide a real number as output.
Square Root Function
A square root function is a function that incorporates a square root, shown as \( y = \sqrt{x} \) in its simplest form. It is characterized by having only non-negative outputs when the radicand is non-negative.For our specific equation \( y = \sqrt{x+5} \), the square root function must take values from its domain and convert them to an output \( y \). Let's list some characteristics of square root functions:
  • They always produce non-negative results. This means for every eligible \( x \), the corresponding \( y \) will either be zero or positive.
  • As the value inside the square root grows larger, \( y \) also increases.
  • At \( x = -5 \), \( y = 0 \), which is also the lowest value \( y \) will ever have.
Square root functions create a type of curve on a graph that starts at a specific point and gradually rises as \( x \) increases, showing a pattern of steady growth.

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

The function given by \(y=0.03 x^{2}+245.50, \quad 0

Find a mathematical model for the verbal statement. For a constant temperature, the pressure \(P\) of a gas is inversely proportional to the volume \(V\) of the gas.

Find a mathematical model for the verbal statement. \(F\) varies directly as \(g\) and inverselv as \(r^{2}\).

Write a sentence using the variation terminology of this section to describe the formula. Volume of a sphere: \(V=\frac{4}{3} \pi r^{3}\)

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=|x-2|, \quad x \leq 2 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.