What Is { (f+g)(x)$} ? ? ? { \begin{align*} f(x) &= -3x^2 + 7x \\ g(x) &= -5x + 2 \end{align*} \} Write Your Answer As A Polynomial Or A Rational Function In Simplest Form. { \square$}$

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Understanding the Concept of $(f+g)(x)$

In mathematics, particularly in the realm of functions, the concept of $(f+g)(x)$ is a fundamental idea that deals with the addition of two functions, $f(x)$ and $g(x)$. This concept is crucial in understanding various mathematical operations and their applications in real-world scenarios.

What is a Function?

Before diving into the concept of $(f+g)(x)$, it's essential to understand what a function is. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. In other words, a function takes an input and produces an output based on a well-defined rule.

What is $(f+g)(x)$?

The expression $(f+g)(x)$ represents the sum of two functions, $f(x)$ and $g(x)$. This means that for a given input $x$, the output of $(f+g)(x)$ is the sum of the outputs of $f(x)$ and $g(x)$.

Given Functions

We are given two functions:

$$\begin{align*} f(x) &= -3x^2 + 7x \\ g(x) &= -5x + 2 \end{align*}$$

Finding $(f+g)(x)$

To find $(f+g)(x)$, we need to add the two functions, $f(x)$ and $g(x)$. This involves combining like terms and simplifying the resulting expression.

Step 1: Add the Coefficients of the $x^2$ Terms

The $x^2$ term in $f(x)$ is $-3x^2$, and there is no $x^2$ term in $g(x)$. Therefore, the $x^2$ term in $(f+g)(x)$ is $-3x^2$.

Step 2: Add the Coefficients of the $x$ Terms

The $x$ term in $f(x)$ is $7x$, and the $x$ term in $g(x)$ is $-5x$. Therefore, the $x$ term in $(f+g)(x)$ is $(7x + (-5x)) = 2x$.

Step 3: Add the Constant Terms

The constant term in $f(x)$ is $0$, and the constant term in $g(x)$ is $2$. Therefore, the constant term in $(f+g)(x)$ is $2$.

Simplifying $(f+g)(x)$

Combining the terms from the previous steps, we get:

$$(f+g)(x) = -3x^2 + 2x + 2$$

This is the simplified form of $(f+g)(x)$.

Conclusion

In conclusion, the expression $(f+g)(x)$ represents the sum of two functions, $f(x)$ and $g(x)$. By adding the two functions, we get a new function, $(f+g)(x)$, which is a polynomial in this case. The resulting function is a simplified form of the original functions, $f(x)$ and $g(x)$.

Applications of $(f+g)(x)$

The concept of $(f+g)(x)$ has various applications in mathematics and real-world scenarios. Some of the applications include:

• Algebra: The concept of $(f+g)(x)$ is used extensively in algebra to simplify expressions and solve equations.
• Calculus: The concept of $(f+g)(x)$ is used in calculus to find the derivative and integral of functions.
• Physics: The concept of $(f+g)(x)$ is used in physics to model real-world phenomena, such as motion and energy.
• Engineering: The concept of $(f+g)(x)$ is used in engineering to design and analyze systems, such as electrical circuits and mechanical systems.

Final Thoughts

In conclusion, the concept of $(f+g)(x)$ is a fundamental idea in mathematics that deals with the addition of two functions. By understanding this concept, we can simplify expressions, solve equations, and model real-world phenomena. The applications of $(f+g)(x)$ are vast and varied, and it is an essential tool for anyone working in mathematics, science, or engineering.

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Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about $(f+g)(x)$.

Q: What is the difference between $(f+g)(x)$ and $f(x) + g(x)$?

A: The expressions $(f+g)(x)$ and $f(x) + g(x)$ are equivalent. The notation $(f+g)(x)$ is often used to emphasize that we are adding two functions, while $f(x) + g(x)$ is a more general notation that can be used for any two expressions.

Q: Can $(f+g)(x)$ be a rational function?

A: Yes, $(f+g)(x)$ can be a rational function. For example, if $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x+1}$, then $(f+g)(x) = \frac{1}{x} + \frac{1}{x+1}$, which is a rational function.

Q: How do I find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both rational functions?

A: To find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both rational functions, you can follow these steps:

1. Find the least common denominator (LCD) of the two rational functions.
2. Add the numerators of the two rational functions, keeping the LCD as the denominator.
3. Simplify the resulting expression.

Q: Can $(f+g)(x)$ be a polynomial?

A: Yes, $(f+g)(x)$ can be a polynomial. For example, if $f(x) = x^2 + 2x$ and $g(x) = 3x^2 - 4x$, then $(f+g)(x) = (x^2 + 2x) + (3x^2 - 4x) = 4x^2 - 2x$, which is a polynomial.

Q: How do I find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both polynomials?

A: To find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both polynomials, you can follow these steps:

1. Combine like terms by adding the coefficients of the same degree terms.
2. Simplify the resulting expression.

Q: Can $(f+g)(x)$ be a constant function?

A: Yes, $(f+g)(x)$ can be a constant function. For example, if $f(x) = 2$ and $g(x) = 3$, then $(f+g)(x) = 2 + 3 = 5$, which is a constant function.

Q: How do I find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both constant functions?

A: To find $(f+g)(x)$ if $f(x)$ and $g(x)$ are both constant functions, you can simply add the two constants.

Conclusion

In conclusion, $(f+g)(x)$ is a fundamental concept in mathematics that deals with the addition of two functions. By understanding this concept, we can simplify expressions, solve equations, and model real-world phenomena. The applications of $(f+g)(x)$ are vast and varied, and it is an essential tool for anyone working in mathematics, science, or engineering.

Final Thoughts

We hope this article has helped you understand the concept of $(f+g)(x)$ and its various applications. If you have any further questions or need additional clarification, please don't hesitate to ask.