/*! 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 61 Sketch the graph of the equation... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 61

Sketch the graph of the equation. $$y=|\sqrt{x}-1|$$

Short Answer

Expert verified
The graph of \( y = |\sqrt{x} - 1| \) is V-shaped, starting at (0,1), going through (1,0), and rising for \( x \geq 1 \).

Step by step solution

01

Identify the Equation Components

The given equation is \( y = |\sqrt{x} - 1| \). This equation involves a square root function \( \sqrt{x} \) and an absolute value function \( |\cdot| \) applied to the expression \( \sqrt{x} - 1 \).
02

Analyze the Square Root Function

The \( \sqrt{x} \) function is defined for \( x \geq 0 \). It takes on values starting from 0 and increases as \( x \) increases. Hence, we only consider \( x \geq 0 \) for our graph.
03

Apply the Absolute Value Function

We rewrite the expression \( |\sqrt{x} - 1| \) in terms of two cases: \( \sqrt{x} - 1 \geq 0 \), which simplifies to \( \sqrt{x} \geq 1 \), and \( \sqrt{x} - 1 < 0 \), which simplifies to \( \sqrt{x} < 1 \).
04

Graph Case 1: \( \sqrt{x} \geq 1 \)

In this case, \( y = \sqrt{x} - 1 \). The graph starts at the point where \( x = 1 \) and continues upwards to the right. At \( x = 1 \), \( y \) is 0, and as \( x \) increases, \( y \) increases linearly.
05

Graph Case 2: \( \sqrt{x} < 1 \)

In this case, \( y = 1 - \sqrt{x} \). The graph starts at the point where \( x = 0 \) and moves to the left. At \( x = 0 \), \( y = 1 \). The graph decreases linearly until it reaches \( y = 0 \) at \( x = 1 \).
06

Combine Both Cases

By combining the two cases, we get a V-shaped graph starting at point (0, 1), descending linearly to (1, 0) for \( x < 1 \), and then ascending linearly from (1, 0) to the right.

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Key Concepts

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

Square Root Functions
Square root functions are essential for understanding various mathematical concepts, especially when graphing equations like the one in this exercise. The basic form is given by the function \( f(x) = \sqrt{x} \). This function is defined only for non-negative values of \( x \), meaning \( x \geq 0 \). Hence, it starts at the origin \((0, 0)\) and extends rightwards as \( x \) increases.

The square root function is characterized by:
  • An increasing function: As \( x \) gets larger, \( \sqrt{x} \) increases.
  • The domain being \([0, \infty)\), where \( 0 \) to infinity represents the range of possible \( x \) values.
  • The range is also \([0, \infty)\), indicating the set of possible output values \( \sqrt{x} \).
When included in an equation with other functions like an absolute value, these properties guide us in understanding the graph's behavior.
Piecewise Functions
Piecewise functions represent a way to define a function using multiple sub-functions, each applying to a certain interval of the main function's domain. In the case of the equation \( y = |\sqrt{x} - 1| \), the absolute value means we need to consider two cases based on the condition \( \sqrt{x} - 1 \geq 0 \) or \( \sqrt{x} - 1 < 0 \).

Here鈥檚 how we break it down:
  • **Case 1:** When \( \sqrt{x} \geq 1 \), the expression becomes \( y = \sqrt{x} - 1 \). It creates a linear part of the graph from the point \( (1, 0) \) extending upwards.
  • **Case 2:** When \( \sqrt{x} < 1 \), the expression becomes \( y = 1 - \sqrt{x} \). This describes the linear segment of the graph from \( (0, 1) \) to \( (1, 0) \).
These two linear cases form a V-shape when plotted, with continuity at the point where the conditions change, i.e., \( x = 1 \). Understanding these divisions is critical for accurate graphing.
Domain and Range Analysis
Analyzing the domain and range is crucial when working with functions like \( y = |\sqrt{x} - 1| \).

**Domain:**
  • This function's domain is influenced by the square root function \( \sqrt{x} \), which restricts \( x \geq 0 \).
  • Thus, the domain is \([0, \infty)\), covering all non-negative numbers.
**Range:**
  • The range is determined by the values \( y \) can take. By analyzing each piece of the piecewise function:
  • For \( \sqrt{x} \geq 1 \), resulting in \( y = \sqrt{x} - 1 \), \( y \) starts from 0 and increases indefinitely.
  • For \( \sqrt{x} < 1 \), resulting in \( y = 1 - \sqrt{x} \), \( y \) ranges from 1 down to 0.
  • Thus, the full range is \([0, 1]\) because these are the only values \( y \) achieves over the defined domain.
Understanding these elements ensures accurate interpretations, helping us to sketch the graph effectively and validate each step while solving related problems.

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Most popular questions from this chapter

(a) Sketch the graph of \(f\) on the given interval \([a, b] .\) (b) Estimate the range of \(f\) on \([a, b] .\) (c) Estimate the intervals on which \(f\) is increasing or is decreasing. $$f(x)=\frac{x^{1 / 3}}{1+x^{4}}$$

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