Problem 1
Question
Fill in the blanks. The ______ of \(f\) and \(g,\) denoted as \(f+g,\) is defined by \((f+g)(x)=\) ______ and the _____ of \(f\) and \(g\) denoted as \(f-g,\) is defined by \((f-g)(x)=\) _____.
Step-by-Step Solution
Verified Answer
The sum of \(f\) and \(g\) is defined by \((f+g)(x) = f(x) + g(x)\) and the difference by \((f-g)(x) = f(x) - g(x)\).
1Step 1: Identify the Operation
The first blank is asking for the type of operation being performed between two functions \(f\) and \(g\). When two functions are being 'added,' it refers to 'sum.' So the answer to the first blank is 'sum.'
2Step 2: Define the Sum of Functions
The second blank involves filling in the result of adding two functions \(f\) and \(g\). The sum of \(f\) and \(g\), denoted by \((f+g)(x)\), is equal to adding the two functions at each point \(x\). The correct expression is \(f(x) + g(x)\). Thus, the second blank is filled with \(f(x) + g(x)\).
3Step 3: Identify the Next Operation
The third blank is again asking for the type of operation, this time indicated by \(f-g\). When functions are being 'subtracted,' it is called the 'difference.' Therefore, the answer to the third blank is 'difference.'
4Step 4: Define the Difference of Functions
The fourth blank focuses on the result of subtracting two functions \(f\) and \(g\). The difference, denoted as \((f-g)(x)\), means subtracting the second function from the first at each point \(x\). It is \(f(x) - g(x)\). So, the fourth blank is filled with \(f(x) - g(x)\).
Key Concepts
Understanding the Sum of FunctionsGrasping the Difference of FunctionsDecoding Function Notation
Understanding the Sum of Functions
When we refer to the **sum of functions**, we are talking about the process of adding two functions together. Imagine you have two functions, \(f(x)\) and \(g(x)\). The sum of these functions is denoted as \((f+g)(x)\). This simply means that for every value of \(x\), you calculate \(f(x)\) and \(g(x)\) separately, and then add these values together.
For example, if \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\), to find \((f+g)(x)\), you would calculate:\[ (f+g)(x) = f(x) + g(x) = (2x + 3) + (x^2 - 1) = x^2 + 2x + 2. \]
This operation is straightforward and involves basic arithmetic addition, applied to function outputs. The significant part here is that you can visualize it as combining or overlaying two separate functions into a new function.
For example, if \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\), to find \((f+g)(x)\), you would calculate:\[ (f+g)(x) = f(x) + g(x) = (2x + 3) + (x^2 - 1) = x^2 + 2x + 2. \]
This operation is straightforward and involves basic arithmetic addition, applied to function outputs. The significant part here is that you can visualize it as combining or overlaying two separate functions into a new function.
Grasping the Difference of Functions
Similar to adding functions, finding the **difference of functions** involves subtracting one function from another. If you have \(f(x)\) and \(g(x)\), the difference is denoted by \((f-g)(x)\), which means subtracting \(g(x)\) from \(f(x)\) for every value of \(x\).
Let's take the same functions as before: \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\). The expression for their difference is:
\[ (f-g)(x) = f(x) - g(x) = (2x + 3) - (x^2 - 1) = -x^2 + 2x + 4. \]
This operation is a simple subtraction, but it's important to pay attention to distributing the subtraction across all terms in \(g(x)\). The result is a new function that represents the difference at each point \(x\). This method is useful in mathematical modeling and comparisons, where you need to assess how two scenarios differ.
Let's take the same functions as before: \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\). The expression for their difference is:
\[ (f-g)(x) = f(x) - g(x) = (2x + 3) - (x^2 - 1) = -x^2 + 2x + 4. \]
This operation is a simple subtraction, but it's important to pay attention to distributing the subtraction across all terms in \(g(x)\). The result is a new function that represents the difference at each point \(x\). This method is useful in mathematical modeling and comparisons, where you need to assess how two scenarios differ.
Decoding Function Notation
**Function notation** is a compact and efficient way to describe mathematical functions. A function, generally written as \(f(x)\), represents the relationship between input values \(x\) and their corresponding output values.
This notation helps us easily understand and denote operations like addition and subtraction. For example, when we write \((f+g)(x)\), it tells us to compute \(f(x)\) and \(g(x)\) separately at each input \(x\), and then apply the operation (addition in this case) between them.
Function notation is impactful because:
This notation helps us easily understand and denote operations like addition and subtraction. For example, when we write \((f+g)(x)\), it tells us to compute \(f(x)\) and \(g(x)\) separately at each input \(x\), and then apply the operation (addition in this case) between them.
Function notation is impactful because:
- It simplifies expressions making them easier to read and interpret
- Clearly indicates what operations to perform
- Makes substituting different expressions or values straightforward
Other exercises in this chapter
Problem 1
Fill in the blanks. An equation with a positive constant base and a variable in its exponent, such as \(3^{2 x}=8,\) is called an _____ equation.
View solution Problem 1
Fill in the blanks. \(f(x)=2^{x}\) and \(f(x)=\left(\frac{1}{4}\right)^{x}\) are examples of______________ functions.
View solution Problem 1
Fill in the blanks. \(f(x)=e^{x}\) is called the natural___ function. The base is _.
View solution Problem 1
Fill in the blanks. The logarithm of a _______, such as \(\log _{3} 4 x,\) equals the sum of the logarithms of the factors.
View solution