Problem 11
Question
For the functions, find \(f(5)\). $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline f(x) & 2.3 & 2.8 & 3.2 & 3.7 & 4.1 & 5.0 & 5.6 & 6.2 \\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
The value of \( f(5) \) is 4.1.
1Step 1: Understand the Table Structure
The table shows values of the function \( f(x) \) for specific values of \( x \). The first row represents the inputs \( x \), and the second row gives the corresponding function values \( f(x) \). Our task is to find \( f(5) \).
2Step 2: Locate the Input Value
Find the column where the value of \( x \) equals 5. This column will provide the corresponding function value for \( f(5) \).
3Step 3: Identify the Function Value
In the column where \( x = 5 \), the corresponding \( f(x) \) value is directly below it. According to the table, \( f(5) = 4.1 \).
4Step 4: Conclusion
The value of \( f(5) \) is 4.1 according to the provided table.
Key Concepts
Mathematical TablesStep-by-Step SolutionFunction Values
Mathematical Tables
Mathematical tables are powerful tools used to quickly find the value of a function at specific input values. Imagine a table with rows and columns. In one row, you have different input values, usually tagged as the "x-row." Below each of these x-values, you find their corresponding function values in the "f(x)-row." This arrangement helps us easily understand the relationship between inputs and outputs of a function.
The primary advantage of mathematical tables is simplicity. They provide a ready reference to look up values without performing calculations. However, tables are mostly helpful for discrete data, where x-values are whole numbers or specific values. When the x-values are closely packed, you might even infer the pattern or trend of the function by examining the sequence of values. In exams or real-world situations, being familiar with how to read these tables can be extremely helpful for solving problems efficiently.
The primary advantage of mathematical tables is simplicity. They provide a ready reference to look up values without performing calculations. However, tables are mostly helpful for discrete data, where x-values are whole numbers or specific values. When the x-values are closely packed, you might even infer the pattern or trend of the function by examining the sequence of values. In exams or real-world situations, being familiar with how to read these tables can be extremely helpful for solving problems efficiently.
Step-by-Step Solution
When tackling a function evaluation problem, it's crucial to follow a systematic approach. A step-by-step solution guides you through the logical process of finding the desired function value.
- **Understand the Table Structure:** First, familiarize yourself with the layout of the table. Identify which row contains the input values (x-values) and which row contains the function values.- **Locate the Input Value:** The next step is finding the specific column where the input value you are interested in is located. In our problem, to find \( f(5) \), look for the column with an x-value of 5.- **Identify the Function Value:** Once the correct column is located, read the function value directly below the input value. The table already provides this pre-calculated function value.- **Conclude with Confidence:** Conclude by stating the function value clearly, as found in the table, ensuring there's no ambiguity. For example, \( f(5) = 4.1 \) in our exercise.
By following such a step-by-step method, you're less likely to make errors, and you'll arrive at an accurate solution quickly.
- **Understand the Table Structure:** First, familiarize yourself with the layout of the table. Identify which row contains the input values (x-values) and which row contains the function values.- **Locate the Input Value:** The next step is finding the specific column where the input value you are interested in is located. In our problem, to find \( f(5) \), look for the column with an x-value of 5.- **Identify the Function Value:** Once the correct column is located, read the function value directly below the input value. The table already provides this pre-calculated function value.- **Conclude with Confidence:** Conclude by stating the function value clearly, as found in the table, ensuring there's no ambiguity. For example, \( f(5) = 4.1 \) in our exercise.
By following such a step-by-step method, you're less likely to make errors, and you'll arrive at an accurate solution quickly.
Function Values
The concept of function values is foundational in mathematics. A function value is what you get when you substitute an input (often denoted as x) into a function. For instance, given a function, you might be asked to find \( f(x) \) for a particular x value, like \( f(5) \).
- **Significance of Function Values:** These values represent outputs of a function for given inputs. They are crucial in understanding how a function behaves and in graphing.- **Real-world Examples:** Think of a function as a machine. You input something, say, the number of hours worked, and out comes the corresponding pay, based on a fixed hourly rate. Every input value in a function has exactly one output—this is known as the uniqueness property of functions.
Function values help you to make predictions, solve equations, and interpret data. By using tools like mathematical tables, you can efficiently retrieve these values, aiding in faster problem-solving and deeper understanding of mathematical relationships.
- **Significance of Function Values:** These values represent outputs of a function for given inputs. They are crucial in understanding how a function behaves and in graphing.- **Real-world Examples:** Think of a function as a machine. You input something, say, the number of hours worked, and out comes the corresponding pay, based on a fixed hourly rate. Every input value in a function has exactly one output—this is known as the uniqueness property of functions.
Function values help you to make predictions, solve equations, and interpret data. By using tools like mathematical tables, you can efficiently retrieve these values, aiding in faster problem-solving and deeper understanding of mathematical relationships.
Other exercises in this chapter
Problem 11
Graph a function \(f(x)\) which is increasing everywhere and concave up for negative \(x\) and concave down for positive \(x\).
View solution Problem 11
A city's population was 30,700 in the year 2000 and is growing by 850 people a year. (a) Give a formula for the city's population, \(P\), as a function of the n
View solution Problem 12
Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\).
View solution Problem 12
A population, currently 200 , is growing at \(5 \%\) per year. (a) Write a formula for the population, \(P\), as a function of time, \(t\), years in the future.
View solution