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Chapter 3: Problem 79

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x+1)-2\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 2 & 4 & 3 & 7 & 8 & 10\end{array}$$

Short Answer

Expert verified
The numerical representation of \(g(x)\) is \(g(1)=2, g(2)=1, g(3)=5, g(4)=6, g(5)=8\).

Step by step solution

01

Understanding the relationship

We are given the relationship between the functions: \(g(x) = f(x+1) - 2\). This means to find \(g(x)\), we need to take the value of \(f\) at \(x+1\) and then subtract 2 from it.
02

Analyzing given data for function \(f\)

The table provides us with specific values of \(f(x)\) for different \(x\): - \(f(1) = 2\)- \(f(2) = 4\)- \(f(3) = 3\)- \(f(4) = 7\)- \(f(5) = 8\)- \(f(6) = 10\).
03

Calculating \(g(1)\)

To find \(g(1)\), use \(f(2)\) because \(g(1) = f(1+1) - 2 = f(2) - 2\). Thus, \(g(1) = 4 - 2 = 2\).
04

Calculating \(g(2)\)

For \(g(2)\), use \(f(3)\): \(g(2) = f(2+1) - 2 = f(3) - 2\). Thus, \(g(2) = 3 - 2 = 1\).
05

Calculating \(g(3)\)

For \(g(3)\), use \(f(4)\): \(g(3) = f(3+1) - 2 = f(4) - 2\). Thus, \(g(3) = 7 - 2 = 5\).
06

Calculating \(g(4)\)

For \(g(4)\), use \(f(5)\): \(g(4) = f(4+1) - 2 = f(5) - 2\). Thus, \(g(4) = 8 - 2 = 6\).
07

Calculating \(g(5)\)

For \(g(5)\), use \(f(6)\): \(g(5) = f(5+1) - 2 = f(6) - 2\). Thus, \(g(5) = 10 - 2 = 8\).
08

Tabulating \(g\) values

Now that we have calculated the values:- \(g(1) = 2\)- \(g(2) = 1\)- \(g(3) = 5\)- \(g(4) = 6\)- \(g(5) = 8\).

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Key Concepts

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

Numerical Representation of Functions
When dealing with functions, a numerical representation helps us understand how a function behaves by showing input-output pairs.
For the functions in our exercise, we use numbers to represent specific points, making it easier to evaluate and manipulate the functions step by step.
For instance, for function \(f(x)\), each \(x\) value has a corresponding \(f(x)\) value:
  • When \(x = 1\), \(f(x) = 2\)
  • When \(x = 2\), \(f(x) = 4\)
This kind of representation gives us a clear and straightforward method to handle functions, allowing you to calculate other function values like \(g(x)\) based on transformations.
Function Tables
A function table organizes values of functions into columns and rows, making numerical data easy to read and compare.
In the provided exercise, the table for function \(f(x)\) signifies how different \(x\) inputs correspond to specific \(f(x)\) values, like:
  • For \(x = 3\), \(f(x) = 3\)
  • For \(x = 4\), \(f(x) = 7\)
Moreover, function tables are crucial for quickly identifying the number needed in transformations or operations, as we do when calculating \(g(x)\). By organizing data systematically, tables provide a visual cue to solve function-related problems efficiently.
Function Operations
Function operations involve manipulating functions to produce new ones.
In our exercise, this involves transforming \(f(x)\) to get \(g(x)\) using the relationship \(g(x) = f(x+1) - 2\).
To perform this:
  • Locate \(f(x+1)\) from the table for each given \(x\)
  • Subtract 2 to find \(g(x)\)
For example, to find \(g(1)\), we see \(f(2) = 4\) and calculate \(g(1) = 4 - 2 = 2\).
This method showcases how simple arithmetic operations can be applied systematically to functions, resulting in transformations or entirely new functions.

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Most popular questions from this chapter

One airline ticket costs \(\$ 250\). For each additional airline ticket sold to a group, the price of every ticket is reduced by \(\$ 2 .\) For example, 2 tickets \(\operatorname{cost} 2 \cdot 248=\$ 496\) and 3 tickets \(\operatorname{cost} 3 \cdot 246=\$ 738\) (a) Write a quadratic function that gives the total cost of buying \(x\) tickets. (b) What is the cost of 5 tickets? (c) How many tickets were sold if the cost is \(\$ 5200 ?\) (d) What number of tickets sold gives the greatest cost?

Use a table to solve. \(x^{2} \geq 3 x+10\)

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Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x)+7\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 5 & 1 & 6 & 2 & 7 & 9\end{array}$$

Exposure When a person breathes carbon monoxide (CO), it enters the bloodstream to form carboxyhemoglobin (COHb), which reduces the transport of oxygen to tissues. The formula given by \(T(x)=0.0078 x^{2}-1.53 x+76\) approximates the number of hours \(T\) that it takes for a person's bloodstream to reach the \(5 \%\) COHb level, where \(x\) is the concentration of \(\mathrm{CO}\) in the air in parts per million (ppm) and \(50 \leq x \leq 100\). (Smokers routinely have a \(5 \%\) concentration.) Estimate the CO concentration \(x\) necessary for a person to reach the \(5 \%\) COHb level in \(4-5\) hours.
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