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Chapter 1: Problem 85

In Exercises 83-86, assume that the domain of \(f\) is the set \(A=\\{-2, -1, 0, 1, 2\\}\). Determine the set of ordered pairs that represents the function \(f\). \(f(x)=|x|+2\)

Short Answer

Expert verified
The ordered pairs representing the function \(f(x) = |x| + 2\) are (-2,4), (-1,3), (0,2), (1,3), (2,4).

Step by step solution

01

Calculate \(-2\)

The first element in our set A is -2. Substitute -2 in place of \(x\) in the equation \(f(x) = |x| + 2\). The equation now becomes \(f(-2) = |-2| + 2\). The absolute value of -2 is 2, so the result is \(f(-2) = 2 + 2 = 4\). Therefore, the first ordered pair is (-2, 4).
02

Calculate \(-1\)

Now substitute -1 in place of \(x\). The equation becomes \(f(-1) = |-1| + 2\). The absolute value of -1 is 1, so \(f(-1) = 1 + 2 = 3\). Therefore, the second ordered pair is (-1, 3).
03

Calculate \(0\)

When \(x = 0\), the equation becomes \(f(0) = |0| + 2\). The absolute value of 0 is 0, so \(f(0) = 0 + 2 = 2\). Therefore, the third ordered pair is (0, 2).
04

Calculate \(1\)

Now substitute 1 in place of \(x\). The equation becomes \(f(1) = |1| + 2\). The absolute value of 1 is 1, so \(f(1) = 1 + 2 = 3\). Therefore, the fourth ordered pair is (1, 3).
05

Calculate \(2\)

When \(x = 2\), the equation becomes \(f(2) = |2| + 2\). The absolute value of 2 is 2, so \(f(2) = 2 + 2 = 4\). Therefore, the fifth and final ordered pair is (2, 4).

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Key Concepts

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

Absolute Value
The concept of absolute value is a key component in understanding this exercise. Absolute value refers to the distance a number is from zero on the number line, regardless of direction. This means it is always non-negative.
For example:
  • The absolute value of -3 is written as \(|-3|\) and equals 3, because -3 is three units away from zero.
  • Similarly, \( |-2|\) is 2, and so forth.
This property becomes crucial when evaluating functions involving absolute values, such as \( f(x) = |x| + 2\). Here, regardless of whether \( x\) is negative or positive, the family of numbers produced will always start at 2, due to the nature of absolute value and the additional planar shift of +2.
Ordered Pairs
Ordered pairs are fundamental in the realm of functions and coordinate geometry. An ordered pair is typically written in the form \( (x, y)\), where \( x\) is the input value and \( y\) is the corresponding output value determined by the function.
To understand how ordered pairs work in functions, consider the function \( f(x) = |x| + 2 \). For each \( x\) value in the domain of the function, calculate the corresponding \( y\), which gives us the set of ordered pairs:
  • When \( x = -2, f(-2) = 4\), so (-2, 4) is an ordered pair.
  • Similarly, for \( x = -1, f(-1) = 3\), resulting in the ordered pair (-1, 3).
  • This pattern continues with (0, 2), (1, 3), and (2, 4).
Ordered pairs provide a clear way to visualize the behavior of a function, especially when plotted on a graph.
Domain of a Function
The domain of a function defines all the possible input values (often \( x\) values) that the function can accept. For the function described in this exercise, the domain is specifically set as \( A = \{-2, -1, 0, 1, 2\}\).
This means we only calculate the function for these points, and determine the outcome based on \( f(x) = |x| + 2\).
By understanding the domain, we can:
  • Clearly know the exact inputs we need to evaluate.
  • Consequently, know all potential outputs based on these inputs.
Domains are vital as they inform us of the limitations or breadth of a function. This sets a framework within which the function operates and provides insights into the relationship between the variable inputs and corresponding outputs.

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Most popular questions from this chapter

An overhead garage door has two springs, one on each side of the door (see figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed.

SPORTS The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 211.3 1976 221.5 1980 218.7 1984 218.5 1988 225.8 1992 213.7 1996 227.7 2000 227.3 2004 229.3 2008 225.8 (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the \(regression\) feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012.

In Exercises 67-74, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(z\) varies directly as the square of \(x\) and inversely as \(y\). (\(z = 6\) when \(x= 6\) and \(y = 4\).)

MEASUREMENT When buying gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Use this information to find a linear model that relates liters \(y\) to gallons \(x\). Then use the model to find the numbers of liters in 5 gallons and 25 gallons.

Statisticians use a measure called ________ of________ ________ to find a model that approximates a set of data most accurately.
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