Function Operations: Videos & Practice Problems

Understanding operations with functions is crucial in algebra. Adding and subtracting functions involves combining like terms, similar to polynomials, while ensuring the domain reflects common values. For multiplication, distribute terms and consider domain restrictions, especially when dividing, as the denominator cannot equal zero. Notation may vary, such as f + g or f(g). Key concepts include polynomial degree, terms, and domain determination, which are essential for simplifying expressions and solving equations effectively.

Adding & Subtracting Functions

In mathematics, understanding how to perform operations on functions, such as addition and subtraction, is essential. When adding or subtracting functions, the process mirrors that of combining polynomials. For instance, if we have two polynomials, we can combine like terms to simplify the expression. For example, if we add the polynomials \( f(x) = x^2 + 4 \) and \( g(x) = 5x + 7 \), we can express this as \( f(x) + g(x) = x^2 + 5x + 11 \). This can also be denoted using the notation \( (f + g)(x) \).

When subtracting functions, the same principles apply. For example, if we have \( g(x) - h(x) \), we distribute the negative sign and combine like terms accordingly. It’s crucial to pay attention to the notation, as it can vary, but the underlying operations remain consistent.

Finding the domain of the resulting function after addition or subtraction involves identifying the common values for the individual functions. For instance, if \( f(x) = \frac{1}{x} \) and \( g(x) = x^2 + x + 2 \), the domain of \( f \) excludes \( x = 0 \) due to the denominator, while \( g \) has a domain of all real numbers. Therefore, the domain of \( f + g \) is also restricted to \( x \neq 0 \).

In another example, if we consider \( g(x) = x^2 + x + 2 \) and \( h(x) = x + \sqrt{x - 8} \), we find that the domain of \( h \) is restricted to \( x \geq 8 \) because the expression under the square root must be non-negative. Consequently, the domain of \( g - h \) will also be \( x \geq 8 \), as this is the only restriction present in the operation.

In summary, when adding or subtracting functions, it is important to combine like terms and carefully determine the domain of the resulting function by considering the restrictions imposed by each individual function. This foundational understanding is crucial for further studies in algebra and calculus.

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 and determine the domain of the resulting expressions.

To find \( f + g \), we combine the two functions: \[f + g = f(x) + g(x) = \left(\sqrt{x + 4} + 30\right) + \left(\sqrt{x + 4} - 2x + 35\right).\]This simplifies to:\[f + g = 2\sqrt{x + 4} - 2x + 65.\]The domain of this function is determined by the square root, which requires that the expression inside must be non-negative. Thus, we set up the inequality:\[x + 4 \geq 0.\]Solving this gives \( x \geq -4 \). Therefore, the domain of \( f + g \) is \( [-4, \infty) \).

Next, we consider \( f - g \):\[f - g = f(x) - g(x) = \left(\sqrt{x + 4} + 30\right) - \left(\sqrt{x + 4} - 2x + 35\right).\]Distributing the negative sign leads to:\[f - g = \sqrt{x + 4} + 30 - \sqrt{x + 4} + 2x - 35.\]The square root terms cancel, resulting in:\[f - g = 2x - 5.\]Although this expression appears to be a linear function with no restrictions, we must still consider the original functions' domains. The square root condition remains, so we again find:\[x + 4 \geq 0 \implies x \geq -4.\]Thus, the domain for \( f - g \) is also \( [-4, \infty) \).

In summary, both the addition and subtraction of the functions yield the same domain of \( [-4, \infty) \), highlighting the importance of considering the original functions when determining the domain, even after simplification.

Multiplying & Dividing Functions

In the study of functions, understanding how to multiply and divide them is essential, as it allows for the exploration of new relationships and behaviors between different mathematical expressions. 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 \( f \cdot g = \sqrt{x} (3x - 6) = 3x\sqrt{x} - 6\sqrt{x} \). The domain of the resulting function is determined by the intersection of the individual domains of \( f \) and \( g \). For \( f(x) \), the domain is \( [0, \infty) \) since the square root function is defined for non-negative values. For \( g(x) \), which is a polynomial, the domain is 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 domain of the quotient is \( [0, 2) \cup (2, \infty) \), allowing all positive values 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 \( x^3 + 2x^2 - 4x - 8 \). Since both functions are polynomials, their combined 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 exclusion of \( x = -2 \) from the domain, resulting in \( (-\infty, -2) \cup (-2, \infty) \) for the quotient.

In summary, when performing operations on functions, it is crucial to consider the domains carefully, especially when division is involved, as restrictions can significantly affect the resulting function's domain. Understanding these principles will enhance your ability to work with functions effectively.

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 are the domain considerations?

Adding and subtracting functions involves combining like terms, similar to polynomials. For example, if $f\left(x\right)=x^2+4$ and $g\left(x\right)=5x+7$, then $f+g\left(x\right)=x^2+5x+11$. Domain considerations involve ensuring that the values do not make any denominator zero or violate conditions like square roots. For instance, if $f\left(x\right)=x^2+\frac{1}{x}$, the domain excludes $x=0$. Understanding these restrictions is crucial for correctly determining the domain of the resulting function.

What is the process for multiplying and dividing functions, and how do you find the domain?

Multiplying functions involves distributing terms, similar to the FOIL method for polynomials. For example, if $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=3x-6$, then $f*g\left(x\right)=3x\sqrt{x}-6\sqrt{x}$. The domain is the intersection of individual domains, considering restrictions like non-negative values for square roots. Dividing functions requires ensuring the denominator is not zero. For $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=3x-6$, $f/g\left(x\right)=\frac{\sqrt{x}}{3x-6}$ excludes $x=2$ from the domain.

What are the notation variations for function operations?

Function operations can be represented in different notations. For addition, $f+g\left(x\right)$ is equivalent to $f\left(x\right)+g\left(x\right)$. Similarly, subtraction can be written as $f-g\left(x\right)$ or $f\left(x\right)-g\left(x\right)$. For multiplication, $f*g\left(x\right)$ is the same as $f\left(x\right)*g\left(x\right)$. Division can be expressed as $\frac{f}{g}\left(x\right)$ or $\frac{f}{g}\left(x\right)$. Recognizing these variations is crucial for correctly interpreting and solving problems involving function operations.

How do you determine the domain of a function after performing operations?

Determining the domain after performing operations on functions involves considering restrictions from individual functions and the resulting function. For addition and subtraction, check for any denominators that cannot be zero or conditions like square roots. For multiplication, the domain is the intersection of individual domains, considering restrictions like non-negative values for square roots. For division, ensure the denominator is not zero, and exclude values that make it zero. For example, if $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=3x-6$, the domain for $f/g\left(x\right)$ excludes $x=2$.