Function Operations: Videos & Practice Problems

Understanding function operations is crucial in mathematics. Adding and subtracting functions involves combining like terms, while multiplying requires distributing terms. The domain of the resulting function is determined by the common restrictions of the original functions. For example, when dividing functions, ensure the denominator is not zero. Notation may vary, such as f + g or f(g). Mastering these concepts enhances problem-solving skills and prepares students for more complex mathematical challenges.

Adding & Subtracting Functions

In mathematics, understanding how to perform operations with functions, such as addition and subtraction, is essential. The process for adding and subtracting functions is similar to that of polynomials, where like terms are combined. For instance, if we have two functions, f(x) = x² + 4 and g(x) = 5x + 7, their sum can be expressed as f(x) + g(x) = x² + 5x + 11. This can also be denoted using the notation (f + g)(x), which signifies the same operation.

When adding or subtracting functions, it is crucial to determine the domain of the resulting function. The domain is defined as the set of all possible input values (x-values) for which the function is defined. For example, if a function has a term with a variable in the denominator, such as 1/x, the domain must exclude any values that make the denominator zero. Thus, for f(x) = x² + 1/x, the domain is all real numbers except x = 0.

To illustrate, consider the addition of two functions: f(x) = x² + 1/x and g(x) = x² + x + 2. When combined, we get f(x) + g(x) = 2x² + x + 2 + 1/x. The domain of this combined function is determined by the more restrictive domain of the individual functions, which in this case is x ≠ 0.

Similarly, when subtracting functions, such as g(x) - h(x), where g(x) = x² + x + 2 and h(x) = x + √(x - 8), we first distribute the negative sign and then combine like terms. The resulting function is x² + 2 - √(x - 8). The domain for this operation is influenced by the square root function, which requires that the expression inside the square root be non-negative. Therefore, we find that x - 8 ≥ 0, leading to the domain x ≥ 8.

In summary, when performing operations with functions, it is important to combine like terms and carefully analyze the domains of the individual functions to determine the domain of the resulting function. This ensures that the operations are valid and that the functions remain defined across their respective domains.

Adding & Subtracting Functions Example 1

In this exploration of function operations, we analyze two functions: \( f(x) = \sqrt{x + 4} + 30 \) and \( g(x) = \sqrt{x + 4} - 2x + 35 \). The goal is to perform addition and subtraction on these functions while determining the domain of the resulting expressions.

To find \( f + g \), we combine the two functions:

\( f + g = f(x) + g(x) = \sqrt{x + 4} + 30 + \left( \sqrt{x + 4} - 2x + 35 \right) \).

By simplifying, we notice that the square root terms combine:

\( f + g = 2\sqrt{x + 4} - 2x + 65 \).

Next, we determine the domain. The only restriction arises from the square root, which requires that the expression inside must be non-negative:

\( x + 4 \geq 0 \) leads to \( x \geq -4 \). Thus, the domain for \( f + g \) is \( [-4, \infty) \).

Now, we consider \( f - g \):

\( f - g = f(x) - g(x) = \sqrt{x + 4} + 30 - \left( \sqrt{x + 4} - 2x + 35 \right) \).

Distributing the negative sign gives us:

\( f - g = \sqrt{x + 4} + 30 - \sqrt{x + 4} + 2x - 35 \).

Upon simplification, the square root terms cancel, resulting in:

\( f - g = 2x - 5 \).

Despite the absence of square roots in the final expression, we must still consider the original functions' domains. The requirement \( x + 4 \geq 0 \) remains, leading to the same domain of \( [-4, \infty) \) for \( f - g \).

In summary, both operations yield the same domain of \( [-4, \infty) \), highlighting the importance of considering the original functions when determining domain restrictions, particularly when square roots are involved.

Multiplying & Dividing Functions

In the study of function operations, multiplying and dividing functions are essential skills that build on the concepts of addition and subtraction. When multiplying two functions, such as \( f(x) = \sqrt{x} \) and \( g(x) = 3x - 6 \), the product is obtained by distributing one function into the other. This results in \( f \cdot g = \sqrt{x}(3x - 6) = 3x\sqrt{x} - 6\sqrt{x} \). Understanding the domain of the resulting function is crucial; it is determined by the intersection of the domains of the individual functions. For \( f(x) \), the domain is \( [0, \infty) \) due to the square root, while \( g(x) \) has a domain of all real numbers, \( (-\infty, \infty) \). Therefore, the domain of the product \( f \cdot g \) is \( [0, \infty) \), as it is the more restrictive of the two domains.

When dividing functions, the process is similar but requires additional caution regarding the denominator. For the same functions, the division is expressed as \( \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{3x - 6} \). Here, the domain of \( f(x) \) remains \( [0, \infty) \), but \( g(x) \) introduces a restriction: the denominator cannot equal zero. Setting \( 3x - 6 = 0 \) gives \( x = 2 \), which must be excluded from the domain. Thus, the overall domain for the division is \( [0, 2) \cup (2, \infty) \), allowing all positive numbers except for 2.

In another example, consider \( f(x) = x^2 - 4 \) and \( g(x) = x + 2 \). The product \( f \cdot g \) can be calculated using the FOIL method, yielding \( f \cdot g = x^3 + 2x^2 - 4x - 8 \). Since both functions are polynomials, their domain is all real numbers, \( (-\infty, \infty) \). However, when dividing \( f(x) \) by \( g(x) \), the restriction arises from the denominator \( x + 2 \), which cannot be zero. This leads to the restriction \( x \neq -2 \), resulting in a domain of \( (-\infty, -2) \cup (-2, \infty) \) for the division operation.

In summary, when performing operations on functions, it is vital to consider both the resulting expressions and their domains. The process of multiplying and dividing functions not only enhances algebraic skills but also deepens the understanding of how functions interact and the importance of domain restrictions in ensuring valid mathematical expressions.

Given the functions $h\left(x\right)=2x^3-4$ and $k\left(x\right)=x^2+2$, find and fully simplify $h\cdot k\left(x\right)$

• A. $h\cdot k\left(x\right)=2\left(x^5+2x^3-2x^2-4\right)$
• B. $h\cdot k\left(x\right)=2x^5-8$
• C. $h\cdot k\left(x\right)=2x^5+4x^3-8$
• D. $h\cdot k\left(x\right)=x^2+4x+4$

Answer: (A)

Given the functions $L\left(x\right)=x-2$ and $M\left(x\right)=x^2$, calculate $\frac{L}{M}\left(5\right)$

• A. $\frac{L}{M}\left(5\right)=\frac{25}{3}$
• B. $\frac{L}{M}\left(5\right)=\frac53$
• C. $\frac{L}{M}\left(5\right)=\frac{3}{25}$
• D. $\frac{L}{M}\left(5\right)=\frac35$

Answer: (C)

Here’s what students ask on this topic:

How do you add and subtract functions, and what is the domain of the resulting function?

Adding and subtracting functions is similar to combining polynomials. For example, if you have two functions, $f\left(x\right)=x^2+4$ and $g\left(x\right)=5x+7$, you add them by combining like terms: $x^2+5x+11$. The domain of the resulting function is the intersection of the domains of the individual functions. For instance, if one function has a restriction like division by zero, this must be considered in the combined domain. Always check for such restrictions to ensure the domain is correctly defined.

What are the steps to multiply and divide functions, and how do you determine the domain?

To multiply functions, distribute terms as you would with polynomials. For example, multiplying $f\left(x\right)=\sqrt{x}$ by $g\left(x\right)=3x-6$ results in $3x\sqrt{x}-6\sqrt{x}$. The domain is restricted to positive numbers due to the square root. For division, divide the functions and ensure the denominator is not zero. For example, dividing $\sqrt{x}$ by $3x-6$ excludes $x=2$ from the domain. Always determine domain restrictions before simplifying expressions.

What is the importance of notation in function operations, and how can it vary?

Notation in function operations is crucial for clarity and understanding. For example, the notation $f+g\left(x\right)$ represents the addition of functions $f$ and $g$. Similarly, $f-g\left(x\right)$ denotes subtraction. For multiplication, $f\cdot g\left(x\right)$ or $fg\left(x\right)$ can be used, while division is shown as $\frac{f}{g}\left(x\right)$. Understanding these notations helps in correctly interpreting and solving problems involving function operations.

How do you find the domain of a function after performing operations like addition, subtraction, multiplication, or division?

To find the domain after performing operations on functions, consider the domains of the individual functions and any new restrictions introduced by the operation. For addition and subtraction, the domain is the intersection of the individual domains. For multiplication, consider any restrictions like square roots, which limit the domain to non-negative numbers. For division, ensure the denominator is not zero, adding restrictions to the domain. For example, dividing $\sqrt{x}$ by $3x-6$ excludes $x=2$ from the domain. Always determine domain restrictions before simplifying expressions to ensure accuracy.

What are some common mistakes to avoid when performing operations on functions?

Common mistakes when performing operations on functions include neglecting domain restrictions, misinterpreting notation, and incorrect simplification. Always check for restrictions like division by zero or negative values under square roots. Be mindful of notation differences, such as $f\cdot g\left(x\right)$ for multiplication and $\frac{f}{g}\left(x\right)$ for division. Simplify expressions carefully, ensuring all terms are combined correctly. By paying attention to these details, you can avoid errors and accurately perform function operations.