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Chapter 4: Problem 11

Decide whether each function as graphed or defined is one-to-one. $$y=\sqrt{36-x^{2}}$$

Short Answer

Expert verified
The function is not one-to-one.

Step by step solution

01

Understanding One-to-One Functions

A function is one-to-one if and only if each output value is associated with exactly one input value. This means that no horizontal line should intersect the graph of the function more than once.
02

Analyze the Given Function

Consider the function defined by the equation: \[ y = \sqrt{36 - x^2} \]This represents the top half of a circle with radius 6 centered at the origin.
03

Check the Horizontal Line Test

Visualize or sketch the graph of the function. The graph will be a semicircle above the x-axis. Draw horizontal lines. If any horizontal line intersects the graph more than once, the function is not one-to-one.
04

Determine the Result

Upon analyzing the graph, any horizontal line drawn within the domain of \( -6 \le x \le 6 \) will intersect the graph at two points, confirming that the function is not one-to-one.

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Key Concepts

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

Horizontal Line Test
The Horizontal Line Test is a quick way to determine if a function is one-to-one. Imagine drawing a series of horizontal lines (parallel to the x-axis) across the graph of the function. If any horizontal line intersects the graph at more than one point, the function is not one-to-one. This is because multiple input values (x-values) would correspond to the same output value (y-value).
To perform this test:
  • Plot the graph of the function.
  • Draw horizontal lines across different parts of the graph.
  • Check for intersections with the graph.
If any horizontal line meets the graph more than once, the function fails the test.
Function Analysis
Analyzing a function involves understanding its properties and behavior over its domain. For the given function \( y = \sqrt{36 - x^2} \), it鈥檚 important to recognize it's restricted to the range of a semicircle. Here's how to analyze the function:
  • Identify the domain and range: \( -6 \le x \le 6\) \(0 \le y \le 6 \).
  • Plot the function: The equation represents the top half of a circle centered at \((0,0) \) with radius 6.
  • Note symmetry: The function is symmetric about the y-axis.
Understanding these properties aids in visualizing how the function behaves and why it might pass or fail the Horizontal Line Test.
Semicircle Graph
A semicircle graph is essentially half of a circle. For this function \( y = \sqrt{36 - x^2} \), we are dealing with the upper half. This form occurs since square roots entail non-negative solutions only. Let鈥檚 break down the semicircle:
The radius is 6, meaning the semicircle spans from \( x = -6 \) to \( x = 6 \). The center at the origin simplifies the understanding of this geometric shape.
  • When \( x = 0 \), y reaches its maximum at 6.
  • When \( x = \pm 6 \), y is 0, laying on the x-axis.
  • The semicircle is symmetric along the y-axis.
By recognizing these features, we can confidently explain why this graph, intersected by any horizontal line more than once within this range, demonstrates that the function is not one-to-one.

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Most popular questions from this chapter

World Population Growth since 2000 , world population in millions closely fits the exponential function $$ y=6084 e^{00120 x} $$ where \(x\) is the number of years since 2000 . (Source: U.S. Census Bureau.) (a) The world population was about 6853 million in 2010 . How closely does the function approximate this value? (b) Use this model to predict the population in 2020 . (c) Use this model to predict the population in 2030 . (d) Explain why this model may not be accurate for 2030 .

(Modeling) Solve each problem. See Example 11 . Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c||c|c} \hline \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ \hline 1000 & 899 & 7000 & 411 \\ \hline 2000 & 795 & 8000 & 357 \\ \hline 3000 & 701 & 9000 & 308 \\ \hline 4000 & 617 & 10,000 & 265 \\ \hline 5000 & 541 & & \\ \hline \end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure \(P\) at altitude \(x\). (b) Would a linear or an exponential function fit the data better? (c) The function $$ P(x)=1013 e^{-0.0001341 x} $$ approximates the data. Use a graphing calculator to graph \(P\) and the data on the same coordinate axes. (d) Use \(P\) to predict the pressures at \(1500 \mathrm{m}\) and \(11,000 \mathrm{m},\) and compare them to the actual values of 846 millibars and 227 millibars, respectively.

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=4 x-5$$

Use a graphing calculator to find the solution set of each equation. Give solutions to the nearest hundredth. $$\log _{10} x=x-2$$

Use the definition of inverses to determine whether \(f\) and \(g\) are inverses. $$f(x)=x^{2}+3, \quad x \geq 0 ; \quad g(x)=\sqrt{x-3}, \quad x \geq 3$$
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