/*! 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 81 each function has a graph with a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 81

each function has a graph with an endpoint (a translation of the point (0,0) .) Enter each into your calculator in an appropriate viewing window, and, using your knowledge of the graph of \(y=\sqrt{x}\), determine the domain and range of the function. (Hint: Locate the endpoint.) $$y=10 \sqrt{x-20}+5$$

Short Answer

Expert verified
Domain: \([20, \infty)\), Range: \([5, \infty)\).

Step by step solution

01

Understand the base function

The base function in question is the square root function, given by \( y = \sqrt{x} \). The graph of this function starts at the point (0,0), and is defined for all \( x \geq 0 \). The graph is only in the first quadrant of the coordinate plane.
02

Analyze the Transformation

The given function is \( y = 10\sqrt{x - 20} + 5 \). This is a transformation of the base function \( y = \sqrt{x} \). The transformation \( x - 20 \) shifts the graph 20 units to the right, and the transformation \( +5 \) shifts the graph 5 units upward. Additionally, the coefficient 10 vertically stretches the graph.
03

Determine the Endpoint

The expression under the square root \( x - 20 \) must be non-negative, meaning \( x \geq 20 \). Thus, the endpoint of the function \( y = 10\sqrt{x - 20} + 5 \) will be at \( x = 20 \). Plugging \( x = 20 \) into the function gives \( y = 10\sqrt{0} + 5 = 5 \). Therefore, the endpoint is at (20, 5).
04

Identify the Domain

Since the square root function \( \sqrt{x - 20} \) is defined for \( x \geq 20 \), the domain of the given function \( y = 10\sqrt{x - 20} + 5 \) is \( x \geq 20 \). In interval notation, this is \([20, \infty)\).
05

Identify the Range

The smallest value of \( y \) occurs at the endpoint where \( y = 5 \). As \( x \) increases beyond \( x = 20 \), the value of \( y = 10\sqrt{x - 20} + 5 \) can increase without bound because the square root function gradually increases. Hence, the range is \( y \geq 5 \). In interval notation, this is \([5, \infty)\).

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.

Function Transformations
Function transformations help us understand how the graph of a function changes with modifications. Essentially, these transformations can alter the position, size, or shape of a graph.
  • **Translation**: A function can be shifted horizontally or vertically. In the function \( y = 10 \sqrt{x - 20} + 5 \), the entire graph is moved 20 units to the right (due to \( x - 20 \)) and 5 units up (due to \( +5 \)). This means the original starting point \((0,0)\) of \( y = \sqrt{x} \) is now at \((20, 5)\).
  • **Scaling**: The coefficient before the square root, 10, is a vertical stretch. It makes the graph steeper, meaning for each value of \( x \), \( y \) increases faster than in the unscaled base function.
Understanding transformations is key because it allows us to predict the graph's new shape and position based on changes to the function's expression. Graphs can be rescaled, mirrored, or translated, all based on the type of transformation applied.
Domain and Range
In any function, the domain and range are critical. The domain refers to all possible input values (or \( x \)-values) for which the function is defined.
  • The domain of \( y = 10\sqrt{x - 20} + 5 \) is based on the expression inside the square root, \( x - 20 \). Since the square root is only defined for non-negative values, \( x \) must be 20 or greater. Thus, the domain is \([20, \infty)\).
The range, on the other hand, refers to all possible output values (or \( y \)-values) the function can produce.
  • Since the function's starting \( y \)-value is 5, found at the endpoint \((20, 5)\), and \( y \) increases as \( x \) increases, the range is \([5, \infty)\). The graph can rise indefinitely beyond 5.
Knowing domain and range helps students understand restrictions and potential outputs the function can take, which is vital when analyzing graphs.
Square Root Function
The square root function, denoted as \( y = \sqrt{x} \), provides a basic framework for understanding other, more complex functions.
  • Its graph starts from the point \((0,0)\), where \( x \geq 0 \), and increases gradually. This creates a distinct curved line in the positive quadrant of the graph.
When modifying the square root function as seen in \( y = 10\sqrt{x - 20} + 5 \), we use this basic shape and adjust it via transformations.
  • The transformation shifts the start to \((20,5)\), and each transformation element alters the way this curve behaves (through translation and stretching).
  • The square root's natural non-linearity means that although the change is subtle at first, \( y \) increases more rapidly as \( x \) goes further from the initial starting point.
Understanding the square root function and its behaviors is foundational for grasping how transformations affect it, leading to proper graphical analysis of transformed functions.

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

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=7-x\) must be less than 0.002 unit from 1.2.

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\sqrt{x}$$

Solve each problem. When a model kite was flown in crosswinds in tests, it attained speeds of 98 to 148 feet per second in winds of 16 to 26 feet per second. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=-2 x^{6}-8 x^{2}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=5 x$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.