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Chapter 8: Problem 62

Use the graph of the functions below to answer Exercises 59 through 70 If \(g(-2)=8,\) write the corresponding ordered pair.

Short Answer

Expert verified
The ordered pair is \((-2, 8)\).

Step by step solution

01

Understand the Given Function Value

We are given that \( g(-2) = 8 \). This means that when \( x = -2 \), the function \( g \) outputs a value of \( 8 \). In other words, at \( x = -2 \), the \( y \)-value is \( 8 \).
02

Write the Ordered Pair

An ordered pair is written in the form \( (x, y) \). Since the problem states \( g(-2) = 8 \), this corresponds to the ordered pair \( (-2, 8) \). Here, \( -2 \) is the \( x \)-coordinate and \( 8 \) is the \( y \)-coordinate.

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Key Concepts

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

Function Evaluation
Function evaluation is a critical concept in algebra and involves finding the output of a function for a specific input value. In this context, a function can be visualized as a machine that takes an input, processes it, and provides an output. When we see notation like \( g(-2) = 8 \), it means that if you "evaluate" the function \( g \) at \( x = -2 \), you get an output of \( 8 \).
This process helps you understand the behavior of the function at specific points.
Function evaluation is crucial for various applications including graphing and solving equations.
Graph Interpretation
Interpreting graphs is a skill that connects the abstract symbols of algebra to visual representation. Graphs provide a snapshot of a function's behavior through the use of Cartesian coordinates. Each point on a graph represents an ordered pair solution to the function.
For instance, if you identify the point \((-2, 8)\) on the graph of the function \( g \), this tells you that when \( x = -2 \), \( y \) is \( 8 \).
  • A point high above the x-axis indicates a large positive value.
  • A point below indicates a negative value.
  • Tracking points horizontally or vertically can lead to various outputs for analysis.
Recognizing patterns and trends within these ordered pairs can offer deep insights into the nature of functions.
Coordinates
Coordinates are fundamental in mathematics to specify positions on a plane using a set of numbers, typically in pairs. In this context, the ordered pair \((-2, 8)\) is part of a coordinate system where \(-2\) is the x-coordinate (horizontal position) and \(8\) is the y-coordinate (vertical position).
Understanding coordinates allows for the precise plotting of points on a graph, aiding in visualization.
  • The x-value shows how far along the horizontal axis a point is.
  • The y-value indicates the vertical distance from the origin.
Coordinates help translate algebraic functions into graph representations, easing understanding of complex mathematical concepts.
Algebraic Functions
Algebraic functions form the backbone of many mathematical studies and are equations involving variables, constants, and arithmetic operations. They are often expressed in the form \( f(x) = y \), where solving for \( y \) at different \( x \) gives specific outputs.
For example, in a scenario where \( g(x) = y \), if \( x = -2 \) results in \( y = 8 \), then \((-2, 8)\) is an outcome of this function. Algebraic functions are used to model real-life situations, providing abstract representations that can be manipulated and studied.
  • Understanding the formula allows you to predict other potential outputs.
  • These functions help in solving equations and inequations.
Grasping algebraic functions increases the ability to navigate different mathematical and real-world challenges.

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Most popular questions from this chapter

The function \(f(x)=\frac{100,000 x}{100-x}\) models the cost in dollars for removing \(x\) percent of the pollutants from a bayou in which a nearby company dumped creosol. a. Find the cost of removing \(20 \%\) of the pollutants from the bayou. (Hint: Find \(f(20) .)\) b. Find the cost of removing \(60 \%\) of the pollutants and then \(80 \%\) of the pollutants. c. Find \(f(90)\), then \(f(95)\), and then \(f(99)\). What happens to the cost as \(x\) approaches \(100 \%\) ?

The total cost (in dollars) for MCD, Inc., Manufacturing Company to produce \(x\) blank audiocassette tapes per week is given by the polynomial function \(C(x)=0.8 x+10,000\). Find the total cost of producing 20,000 tapes per week.

Calculating body-mass index \((B M I)\) is a way to gauge whether a person should lose weight. Doctors recommend that body-mass index values fall between 19 and \(25 .\) The formula for body-mass index \(B\) is \(B=\frac{705 w}{h^{2}},\) where \(w\) is weight in pounds and \(h\) is height in inches. A doctor recorded a body-mass index of 47 on a patient's chart. Later, a nurse notices that the doctor recorded the patient's weight as 240 pounds but neglected to record the patient's height. Explain how the nurse can use the information from the chart to find the patient's height. Then find the height.

In your own words, explain how to find the domain of a function given its graph.

The total revenues (in dollars) for MCD, Inc., Manufacturing Company to sell \(x\) blank audiocassette tapes per week is given by the polynomial function \(R(x)=2 x .\) Find the total revenue from selling 20,000 tapes per week.
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