Problem 81
Question
Use the tables of values for functions \(f\) and \(g\) to find each of the following. a. \((f+g)(1)\) b. \((f-g)(5)\) c. \((f \cdot g)(1)\) d. \((g / f)(5)\) $$ \begin{array}{|c|c|c|c|} \hline x & f(x) & x & g(x) \\ \hline 1 & 3 & 1 & 4 \\ 5 & 8 & 5 & 0 \\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
a. 7, b. 8, c. 12, d. 0
1Step 1: Identify the Values for (f+g)(1)
To find \((f+g)(1)\), we need to determine the values of \(f(1)\) and \(g(1)\) from the tables. According to the tables, \(f(1) = 3\) and \(g(1) = 4\).
2Step 2: Calculate (f+g)(1)
Adding the corresponding values, \[(f+g)(1) = f(1) + g(1) = 3 + 4 = 7\]
3Step 3: Identify the Values for (f-g)(5)
To find \((f-g)(5)\), we need to determine the values of \(f(5)\) and \(g(5)\) from the tables. According to the tables, \(f(5) = 8\) and \(g(5) = 0\).
4Step 4: Calculate (f-g)(5)
Subtract the value of \(g(5)\) from \(f(5)\), \[(f-g)(5) = f(5) - g(5) = 8 - 0 = 8\]
5Step 5: Identify the Values for (f·g)(1)
To find \((f \cdot g)(1)\), we need to determine the values of \(f(1)\) and \(g(1)\) from the tables. According to the tables, \(f(1) = 3\) and \(g(1) = 4\).
6Step 6: Calculate (f·g)(1)
Multiply the corresponding values, \[(f \cdot g)(1) = f(1) \cdot g(1) = 3 \cdot 4 = 12\]
7Step 7: Identify the Values for (g/f)(5)
To find \((g/f)(5)\), we need to determine the values of \(g(5)\) and \(f(5)\) from the tables. According to the tables, \(g(5) = 0\) and \(f(5) = 8\).
8Step 8: Calculate (g/f)(5)
Divide the value of \(g(5)\) by \(f(5)\), \[(g/f)(5) = \frac{g(5)}{f(5)} = \frac{0}{8} = 0\]
Key Concepts
Addition of FunctionsSubtraction of FunctionsMultiplication of FunctionsDivision of Functions
Addition of Functions
When you add two functions, you're essentially adding their values for a given input. Think of it like combining the results of two separate recipes into one delicious dish. To perform the addition of two functions, say \( f(x) \) and \( g(x) \), you simply sum their outputs wherever you decide to measure them. So, \( (f+g)(x) = f(x) + g(x) \).
In the context of the exercise, to find \((f+g)(1)\), we look at \( f(1) \) and \( g(1) \) from the given table of values. In the table, we see that \( f(1) = 3 \) and \( g(1) = 4 \). Adding these values, we get:
\[ (f+g)(1) = f(1) + g(1) = 3 + 4 = 7 \] This result tells us the total "combined output" of both functions at \( x=1 \). By learning how to sum functions, you gain the power to predict and analyze combined effects in various scenarios, from math to physics and economics.
In the context of the exercise, to find \((f+g)(1)\), we look at \( f(1) \) and \( g(1) \) from the given table of values. In the table, we see that \( f(1) = 3 \) and \( g(1) = 4 \). Adding these values, we get:
\[ (f+g)(1) = f(1) + g(1) = 3 + 4 = 7 \] This result tells us the total "combined output" of both functions at \( x=1 \). By learning how to sum functions, you gain the power to predict and analyze combined effects in various scenarios, from math to physics and economics.
Subtraction of Functions
Subtraction of functions works similarly to addition, with one key difference: instead of adding, you subtract. The notion is straightforward: to find the difference in outputs from two functions at any point \( x \), subtract \( g(x) \) from \( f(x) \), represented by \( (f-g)(x) = f(x) - g(x) \). This process allows you to compare variations between two data sets and identify gaps or changes.
Let’s apply it to the exercise to determine \((f-g)(5)\). Our table shows \( f(5) = 8 \) and \( g(5) = 0 \). By subtracting, the computation is as follows:
\[ (f-g)(5) = f(5) - g(5) = 8 - 0 = 8 \] This indicates that at \( x=5 \), the function \( f \) is eight units greater than the function \( g \). You can use subtraction of functions to highlight differences or monitor changes over intervals in your studies.
Let’s apply it to the exercise to determine \((f-g)(5)\). Our table shows \( f(5) = 8 \) and \( g(5) = 0 \). By subtracting, the computation is as follows:
\[ (f-g)(5) = f(5) - g(5) = 8 - 0 = 8 \] This indicates that at \( x=5 \), the function \( f \) is eight units greater than the function \( g \). You can use subtraction of functions to highlight differences or monitor changes over intervals in your studies.
Multiplication of Functions
Combining functions through multiplication involves multiplying their values at a chosen point. This process can take individual outcomes and calculate a compounded result. Using two functions, \( f(x) \) and \( g(x) \), the multiplication is expressed as \( (f \cdot g)(x) = f(x) \cdot g(x) \).
To solve the problem in the exercise for \((f \cdot g)(1)\), we refer to the table to find \( f(1) \) and \( g(1) \), where \( f(1) = 3 \) and \( g(1) = 4 \). Thus, we calculate:
\[ (f \cdot g)(1) = f(1) \cdot g(1) = 3 \cdot 4 = 12 \] This result showcases the amplified effect of outcomes at \( x=1 \). Understanding multiplication of functions is crucial for tasks where outcomes are interdependent, such as calculating areas or solving complex equations.
To solve the problem in the exercise for \((f \cdot g)(1)\), we refer to the table to find \( f(1) \) and \( g(1) \), where \( f(1) = 3 \) and \( g(1) = 4 \). Thus, we calculate:
\[ (f \cdot g)(1) = f(1) \cdot g(1) = 3 \cdot 4 = 12 \] This result showcases the amplified effect of outcomes at \( x=1 \). Understanding multiplication of functions is crucial for tasks where outcomes are interdependent, such as calculating areas or solving complex equations.
Division of Functions
Division of functions involves dividing their outputs at a particular point. Carefully handling division is important to avoid errors, especially since dividing by zero is undefined. The division is represented as \( (g/f)(x) = \frac{g(x)}{f(x)} \), showing how one function compares to another in magnitude.
Our exercise requires finding \((g/f)(5)\). According to the tables, \( g(5) = 0 \) and \( f(5) = 8 \). By dividing, we proceed with calculating:
\[ (g/f)(5) = \frac{g(5)}{f(5)} = \frac{0}{8} = 0 \] This indicates that at \( x=5 \), the output of \( g \) is entirely consumed by \( f \), resulting in zero. Function division is vital in contexts like calculating rates, density, or efficiency in real-world applications. Remember, always ensure the denominator is non-zero to keep calculations valid.
Our exercise requires finding \((g/f)(5)\). According to the tables, \( g(5) = 0 \) and \( f(5) = 8 \). By dividing, we proceed with calculating:
\[ (g/f)(5) = \frac{g(5)}{f(5)} = \frac{0}{8} = 0 \] This indicates that at \( x=5 \), the output of \( g \) is entirely consumed by \( f \), resulting in zero. Function division is vital in contexts like calculating rates, density, or efficiency in real-world applications. Remember, always ensure the denominator is non-zero to keep calculations valid.
Other exercises in this chapter
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