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Chapter 6: Problem 13

A linear function is given. Evaluate the function at the indicated values and then graph the function over its given domain. $$P(x)=\frac{1}{6}, x=0,1,2, \ldots, 6 ; P(2), P(5)$$

Short Answer

Expert verified
Function is constant, \( P(x)=\frac{1}{6} \) for all \( x \).

Step by step solution

01

Understand the linear function

The function given is a constant linear function, \( P(x) = \frac{1}{6} \). This means for any \( x \) value, \( P(x) \) will always be the same, \( \frac{1}{6} \), regardless of \( x \).
02

Evaluate function at specific values

We need to evaluate \( P(x) \) for \( x = 2 \) and \( x = 5 \). \[ P(2) = \frac{1}{6} \] \[ P(5) = \frac{1}{6} \] Since the function is constant, \( P(x) = \frac{1}{6} \) for any \( x \).
03

Graph the function

To graph \( P(x) = \frac{1}{6} \) over the domain \( x = 0, 1, 2, \ldots, 6 \), draw a horizontal line at \( y = \frac{1}{6} \). The x-values range from 0 to 6 along the x-axis, and for each of these, \( y \) will be \( \frac{1}{6} \).
04

Describe the graph

On the graph, you will see a horizontal line intersecting the y-axis at \( y = \frac{1}{6} \). The line is straight and parallel to the x-axis, indicating the function is constant across the domain.

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Key Concepts

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

Constant Functions
A constant function is a special type of linear function where the output value, or y-value, remains the same regardless of the input value, or x-value. In the case of the function \( P(x) = \frac{1}{6} \), no matter what value of \( x \) you substitute into the function, \( P(x) \) will always equal \( \frac{1}{6} \).
This consistency is why it's called a "constant" function, because the result does not change. Constant functions are typically represented graphically as horizontal lines.
  • The equation of a constant function has the form \( f(x) = c \), where \( c \) is a constant.
  • The slope of a constant function is zero because there is no change in y-values as \( x \) changes.
Understanding constant functions helps simplify many problems, as it offers predictable outcomes at any point along the x-axis.
Function Evaluation
Function evaluation is the process of finding the output value of a function for specific input values. In simpler terms, it means finding \( f(a) \) for some \( a \). For the constant function \( P(x) = \frac{1}{6} \), evaluating the function involves substituting a value in for \( x \) and finding the corresponding \( y \).
Since \( P(x) \) is constant, when evaluating \( P(2) \) or \( P(5) \), the process is straightforward:
  • Substitute the given \( x \)-value into the function.
  • In this case, for any \( x \), \( P(x) \) is always \( \frac{1}{6} \).
  • So, \( P(2) = \frac{1}{6} \) and \( P(5) = \frac{1}{6} \).
This makes constant functions particularly simple to evaluate compared to other functions, where outputs can vary with different inputs.
Graph Description
When graphing a constant function like \( P(x) = \frac{1}{6} \), the result is a horizontal line. This horizontal line indicates that the function's value is fixed as \( x \) changes. The line crosses the y-axis at \( y = \frac{1}{6} \) and doesn鈥檛 deviate.
For the domain \( x = 0, 1, 2, \ldots, 6 \), draw the horizontal line at \( y = \frac{1}{6} \) between these x-values. Here's what you can observe:
  • All points on the line will have the y-coordinate \( y = \frac{1}{6} \).
  • The line is parallel to the x-axis.
  • It visually represents that regardless of the x-value, the function's output remains constant.
Graphically understanding constant functions aids in grasping other linear function behaviors as well.
Domain of a Function
The domain of a function is the complete set of possible input values \( x \) that the function can accept. For the function \( P(x) = \frac{1}{6} \), the domain is explicitly given as \( x = 0, 1, 2, \ldots, 6 \).
Hence, this domain includes all integer values from 0 through 6. Each of these x-values, when plugged into the function, results in the same y-value \( \frac{1}{6} \).
The domain helps define the scope of the graphable line on a coordinate plane. Understanding the domain is key to interpreting which x-values can be used in the function, ensuring correct solutions and graphs.
  • It defines the limits over which the function exists or is visually represented.
  • In this example, the domain restricts x to integer values between 0 and 6.
Selecting correct domain values is crucial for accurate function analysis and representation.

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Most popular questions from this chapter

Make a table of function values using the given discrete domain values. Write the values as ordered pairs and then graph the function. $$f(x)=-4+0.8 x, \quad x=5,6,7,8,9,10$$

A piece wise defined function is given. Evaluate the function at the indicated values and then graph the function over its given domain. \(P(x)=\left\\{\begin{array}{ll}1.6 x, & x=0,1,2,3,4,5 \\ 8, & x=6,7,8 \\ 0, & \text { otherwise }\end{array} \quad ; P(5), P(8)\right.\)

Consider the following scenario: The Pick-Chick restaurant charges \(\$ 2.50\) per chicken piece sold on the first 5 pieces and \(\$ 2.00\) per piece thereafter up to 10 pieces. The cost of the chicken is expressed by the piece wise defined function \(f(x)=\left\\{\begin{array}{cl}2.50 x, & x=1,2,3,4,5 \\\ 2.5+2 x, & x=6,7,8,9,10\end{array}\right.\) where \(x\) represents the number of pieces of chicken sold and \(f(x)\) represents cost in dollars. Evaluate \(f(8)\) and interpret the result.

Consider the following scenario: General practitioner (GP) is becoming a popular choice of career for residents in Norway. The annual number of practicing GPs in Norway from 2003 to 2013 can be expressed using the function $$P(x)=310+9.9 x, \quad x=3,4, \ldots, 13$$ where \(x\) represents the number of years since 2000 and \(P(x)\) represents the number of GPs per 100,000 Norwegians. Use this function to answer the following questions. Did the number of practicing GPs per 100,000 Norwegians increase or decrease from 2006 to 2012?

Consider the following scenario: A recent point for discussion has been an increase in the national minimum wage. Joe Roman of the Greater Cleveland Partnership claims that an increase in the minimum wage to \(\$ 15\) per hour in Cleveland would be the "most aggressive minimum wage increase in the country". The annual monthly minimum-wage earnings in the United States from 2007 to 2015 can be modeled by the function $$f(x)=419.9+41.2 x, \quad x=7,8, \ldots, 15$$ where \(x\) represents the number of years since 2000 and \(f(x)\) represents the monthly minimum-wage earning in dollars. Use this function to answer the following questions. Make a table of values for \(x\) and \(f(x)\) using the given domain values. Write the values as ordered pairs.
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