Mastering Function Operations: F(x) And G(x)

Welcome, math enthusiasts! Today, we're diving into the exciting world of function operations. Specifically, we'll be working with two functions, $f(x) = x - 7$ and $g(x) = 7x + 8$, and exploring how to combine them through addition, subtraction, multiplication, and division. Understanding these operations is fundamental to grasping more complex mathematical concepts and is a crucial skill for anyone looking to excel in algebra and beyond. We'll break down each operation step-by-step, ensuring clarity and providing you with the confidence to tackle similar problems on your own. So, grab your notebooks, and let's embark on this mathematical journey together! We'll cover:

• Addition of Functions: $(f+g)(x)$
• Subtraction of Functions: $(f-g)(x)$
• Multiplication of Functions: $(f \cdot g)(x)$
• Division of Functions: $\left(\frac{f}{g}\right)(x)$

By the end of this article, you'll be a pro at performing these basic function operations, setting a strong foundation for your future mathematical endeavors.

Addition of Functions: $(f+g)(x)$

Let's start with the simplest operation: addition of functions. When we talk about $(f+g)(x)$, we're essentially saying we want to add the output of function $f(x)$ to the output of function $g(x)$. In simpler terms, we take the expression for $f(x)$ and the expression for $g(x)$ and combine them by adding them together. Our functions are $f(x) = x - 7$ and $g(x) = 7x + 8$. To find $(f+g)(x)$, we will literally add these two expressions.

We begin by writing out the operation:

$(f+g)(x) = f(x) + g(x)$

Now, substitute the given expressions for $f(x)$ and $g(x)$ into this equation:

$(f+g)(x) = (x - 7) + (7x + 8)$

The next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In our case, the like terms are the 'x' terms ($x$ and $7x$) and the constant terms (-7 and +8).

First, let's group the like terms together. It often helps to write it out like this:

$(f+g)(x) = x + 7x - 7 + 8$

Now, combine the 'x' terms: $x + 7x$. Remember that $x$ is the same as $1x$. So, $1x + 7x = 8x$.

Next, combine the constant terms: $-7 + 8$. This equals $1$.

Putting it all together, we get our final simplified expression for $(f+g)(x)$:

$(f+g)(x) = 8x + 1$

Congratulations! You've successfully found the sum of two functions. This process involves a straightforward substitution and then the basic algebraic skill of combining like terms. It's a fundamental building block for more advanced function manipulation. Remember, the key is to treat $f(x)$ and $g(x)$ as individual expressions that you can then combine according to the specified operation. In addition, we simply add the corresponding terms from each function.

Subtraction of Functions: $(f-g)(x)$

Moving on to subtraction of functions, denoted as $(f-g)(x)$, we perform a similar process to addition, but with a crucial difference: we subtract the entire expression of $g(x)$ from $f(x)$. This means we need to be careful with the signs, especially when $g(x)$ has multiple terms. The operation $(f-g)(x)$ means $f(x) - g(x)$. Our functions remain $f(x) = x - 7$ and $g(x) = 7x + 8$.

Let's set up the equation for subtraction:

$(f-g)(x) = f(x) - g(x)$

Now, substitute the expressions for $f(x)$ and $g(x)$. Here's where the caution comes in: when you subtract $g(x)$, you must subtract each term within $g(x)$. To do this correctly, we enclose $g(x)$ in parentheses:

$(f-g)(x) = (x - 7) - (7x + 8)$

The next vital step is to distribute the negative sign to each term inside the parentheses of $g(x)$. This is because subtracting a quantity is the same as adding its opposite. So, $-(7x + 8)$ becomes $-7x - 8$.

$(f-g)(x) = x - 7 - 7x - 8$

Now, just like with addition, we combine like terms. We group the 'x' terms and the constant terms:

$(f-g)(x) = (x - 7x) + (-7 - 8)$

Combine the 'x' terms: $x - 7x$. Remember, $x$ is $1x$. So, $1x - 7x = -6x$.

Combine the constant terms: $-7 - 8$. This equals $-15$.

Putting it all together, we get the simplified expression for $(f-g)(x)$:

$(f-g)(x) = -6x - 15$

It's important to double-check your signs when performing subtraction. A common mistake is forgetting to distribute the negative sign to all terms within the subtrahend ($g(x)$ in this case). By using parentheses and carefully distributing, you ensure accuracy. This skill is critical as you move on to more complex algebraic manipulations involving polynomials and other expressions. Remember the distributive property of negation: $- (a + b) = -a - b$. Applying this correctly is key to mastering function subtraction.

Multiplication of Functions: $(f \cdot g)(x)$

Now, let's tackle multiplication of functions, denoted as $(f \cdot g)(x)$. This operation involves multiplying the expression for $f(x)$ by the expression for $g(x)$. For our functions, $f(x) = x - 7$ and $g(x) = 7x + 8$. The operation $(f colcdot g)(x)$ means $f(x) \cdot g(x)$.

We start by writing out the operation:

$(f \cdot g)(x) = f(x) \cdot g(x)$

Substitute the expressions for $f(x)$ and $g(x)$:

$(f \cdot g)(x) = (x - 7) \cdot (7x + 8)$

Since both $f(x)$ and $g(x)$ are binomials (expressions with two terms), we will use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to multiply them. This means each term in the first binomial must be multiplied by each term in the second binomial.

1. First terms: Multiply the first term of $(x - 7)$ by the first term of $(7x + 8)$. $x \cdot 7x = 7x^2$

2. Outer terms: Multiply the outer terms of the two binomials. $x \cdot 8 = 8x$

3. Inner terms: Multiply the inner terms of the two binomials. $-7 \cdot 7x = -49x$

4. Last terms: Multiply the last term of $(x - 7)$ by the last term of $(7x + 8)$. $-7 \cdot 8 = -56$

Now, we add these four results together:

$(f \cdot g)(x) = 7x^2 + 8x - 49x - 56$

As with addition and subtraction, the final step is to simplify by combining like terms. The like terms here are $8x$ and $-49x$.

Combine the 'x' terms: $8x - 49x = -41x$.

So, the simplified expression for $(f \cdot g)(x)$ is:

$(f \cdot g)(x) = 7x^2 - 41x - 56$

Mastering multiplication of polynomials is a key skill in algebra. The FOIL method is a systematic way to ensure you don't miss any term pairs. When you multiply functions, you're essentially multiplying their algebraic representations. This operation often results in a polynomial of a higher degree than the original functions, as seen here where we obtained a quadratic expression ($7x^2 - 41x - 56$). Practice this method with different types of polynomials – binomials, trinomials, etc. – to build your proficiency. Remember, accuracy in each multiplication step and careful combining of like terms will lead you to the correct answer every time.

Division of Functions: $\left(\frac{f}{g}\right)(x)$

Finally, let's explore division of functions, represented as $\left(\frac{f}{g}\right)(x)$. This operation means we divide the expression for $f(x)$ by the expression for $g(x)$. Our functions are $f(x) = x - 7$ and $g(x) = 7x + 8$. The operation $\left(\frac{f}{g}\right)(x)$ translates to $\frac{f(x)}{g(x)}$.

Set up the division:

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Substitute the expressions for $f(x)$ and $g(x)$:

$\left(\frac{f}{g}\right)(x) = \frac{x - 7}{7x + 8}$

For this particular problem, the expression $\frac{x - 7}{7x + 8}$ is already in its simplest form. Unlike addition, subtraction, and multiplication, division of functions often results in a rational expression (a fraction where the numerator and denominator are polynomials). There isn't a general simplification method like FOIL for division that applies to all cases. We can only simplify further if there are common factors between the numerator and the denominator, which is not the case here.

An important consideration with division is the domain of the resulting function. The denominator of a fraction cannot be zero. Therefore, for $\left(\frac{f}{g}\right)(x)$ to be defined, $g(x)$ cannot equal zero. We must find the values of $x$ for which $g(x) = 0$ and exclude them from the domain.

Set $g(x)$ equal to zero and solve for $x$:

$7x + 8 = 0$

Subtract 8 from both sides:

$7x = -8$

Divide by 7:

$x = -\frac{8}{7}$

So, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x = -\frac{8}{7}$. We can express this as: $x \in \mathbb{R}, x \neq -\frac{8}{7}$.

Thus, the function is:

$\left(\frac{f}{g}\right)(x) = \frac{x - 7}{7x + 8}$, with the restriction $x \neq -\frac{8}{7}$.

Understanding rational expressions and their domains is a critical part of working with function division. Always remember to check if the denominator can be zero and state any restrictions on the variable $x$. While sometimes the resulting fraction can be simplified by canceling common factors, often it remains as it is. This operation highlights the importance of considering the conditions under which a function is valid.

Conclusion

We've successfully navigated through the four basic operations of functions: addition, subtraction, multiplication, and division, using $f(x) = x - 7$ and $g(x) = 7x + 8$. We found:

• $(f+g)(x) = 8x + 1$
• $(f-g)(x) = -6x - 15$
• $(f \cdot g)(x) = 7x^2 - 41x - 56$
• $\left(\frac{f}{g}\right)(x) = \frac{x - 7}{7x + 8}$, where $x \neq -\frac{8}{7}$

Mastering these operations is a significant step in your journey through mathematics. Each operation builds upon fundamental algebraic skills like combining like terms and the distributive property. Remember to pay close attention to signs during subtraction and to properly distribute during multiplication. For division, always consider the domain restrictions to ensure the function is well-defined.

These concepts are foundational for understanding more complex topics in calculus, pre-calculus, and beyond. Keep practicing these types of problems, and don't hesitate to explore further.

For more in-depth learning on function operations, you can visit Khan Academy's comprehensive section on function operations. They offer excellent tutorials and practice exercises that can further solidify your understanding.

Also, check out Paul's Online Math Notes for detailed explanations and worked examples on various algebra and calculus topics, including function operations.