Given The Function:$\[ F(x) = 7\left(x^4 + 2\right) \\]Evaluate Or Simplify As Needed.

Evaluating and Simplifying the Function $f(x) = 7\left(x^4 + 2\right)$

In mathematics, functions are a fundamental concept that helps us describe relationships between variables. Evaluating and simplifying functions is an essential skill that is used in various mathematical operations, including algebra, calculus, and more. In this article, we will focus on evaluating and simplifying the function $f(x) = 7\left(x^4 + 2\right)$.

Understanding the Function

The given function is $f(x) = 7\left(x^4 + 2\right)$. This function consists of two main components: the variable $x$ and the constant $7$. The variable $x$ is raised to the power of $4$, and then the result is added to $2$. The entire expression is then multiplied by the constant $7$.

Evaluating the Function

To evaluate the function, we need to substitute a value for $x$ and then simplify the expression. Let's consider a specific value for $x$, say $x = 2$. Substituting this value into the function, we get:

$f(2) = 7\left(2^4 + 2\right)$

Using the order of operations, we first evaluate the exponentiation:

$2^4 = 16$

Now, we can substitute this value back into the function:

$f(2) = 7\left(16 + 2\right)$

Next, we add the two numbers inside the parentheses:

$16 + 2 = 18$

Now, we can multiply the result by $7$:

$f(2) = 7 \times 18$

Finally, we can evaluate the multiplication:

$f(2) = 126$

Simplifying the Function

In addition to evaluating the function, we can also simplify it by factoring out common terms. Let's consider the function $f(x) = 7\left(x^4 + 2\right)$. We can factor out the common term $x^4$ from the expression inside the parentheses:

$f(x) = 7\left(x^4 + 2\right) = 7x^4 + 14$

This simplified form of the function is easier to work with and can be used in various mathematical operations.

Graphing the Function

To visualize the function, we can graph it using a coordinate plane. The graph of the function $f(x) = 7\left(x^4 + 2\right)$ is a curve that opens upwards. The graph has a minimum point at $x = 0$, and it increases rapidly as $x$ increases.

Real-World Applications

The function $f(x) = 7\left(x^4 + 2\right)$ has several real-world applications. For example, it can be used to model population growth, where the variable $x$ represents the number of individuals and the function $f(x)$ represents the population size. It can also be used to model economic growth, where the variable $x$ represents the number of years and the function $f(x)$ represents the economic growth rate.

In conclusion, evaluating and simplifying the function $f(x) = 7\left(x^4 + 2\right)$ is an essential skill that is used in various mathematical operations. By understanding the function and its components, we can evaluate and simplify it using various techniques. The function has several real-world applications, including modeling population growth and economic growth. By mastering the skills of evaluating and simplifying functions, we can solve complex mathematical problems and make informed decisions in various fields.

For more information on evaluating and simplifying functions, check out the following resources:

• Khan Academy: Evaluating and Simplifying Functions
• Mathway: Evaluating and Simplifying Functions
• Wolfram Alpha: Evaluating and Simplifying Functions

Q: What is the value of $f(2)$? A: The value of $f(2)$ is $126$.

Q: Can the function $f(x) = 7\left(x^4 + 2\right)$ be simplified? A: Yes, the function can be simplified by factoring out the common term $x^4$.

Q: What are the real-world applications of the function $f(x) = 7\left(x^4 + 2\right)$? A: The function has several real-world applications, including modeling population growth and economic growth.
Evaluating and Simplifying the Function $f(x) = 7\left(x^4 + 2\right)$: Q&A

In our previous article, we discussed evaluating and simplifying the function $f(x) = 7\left(x^4 + 2\right)$. In this article, we will provide a Q&A section to address some of the common questions and concerns that readers may have.

Q: What is the value of $f(0)$? A: To find the value of $f(0)$, we need to substitute $x = 0$ into the function. This gives us:

$f(0) = 7\left(0^4 + 2\right)$

Using the order of operations, we first evaluate the exponentiation:

$0^4 = 0$

Now, we can substitute this value back into the function:

$f(0) = 7\left(0 + 2\right)$

Next, we add the two numbers inside the parentheses:

$0 + 2 = 2$

Now, we can multiply the result by $7$:

$f(0) = 7 \times 2$

Finally, we can evaluate the multiplication:

$f(0) = 14$

Q: Can the function $f(x) = 7\left(x^4 + 2\right)$ be used to model population growth? A: Yes, the function can be used to model population growth. In this case, the variable $x$ represents the number of years, and the function $f(x)$ represents the population size.

Q: How can I graph the function $f(x) = 7\left(x^4 + 2\right)$? A: To graph the function, you can use a coordinate plane and plot the points $(x, f(x))$ for various values of $x$. You can also use a graphing calculator or software to graph the function.

Q: What is the domain of the function $f(x) = 7\left(x^4 + 2\right)$? A: The domain of the function is all real numbers, since there are no restrictions on the values of $x$.

Q: Can the function $f(x) = 7\left(x^4 + 2\right)$ be used to model economic growth? A: Yes, the function can be used to model economic growth. In this case, the variable $x$ represents the number of years, and the function $f(x)$ represents the economic growth rate.

Q: How can I simplify the function $f(x) = 7\left(x^4 + 2\right)$? A: You can simplify the function by factoring out the common term $x^4$ from the expression inside the parentheses. This gives us:

$f(x) = 7\left(x^4 + 2\right) = 7x^4 + 14$

Q: What are some real-world applications of the function $f(x) = 7\left(x^4 + 2\right)$? A: Some real-world applications of the function include modeling population growth, economic growth, and other types of growth or change.

In conclusion, the function $f(x) = 7\left(x^4 + 2\right)$ is a useful tool for modeling various types of growth or change. By understanding the function and its components, we can evaluate and simplify it using various techniques. We hope that this Q&A section has been helpful in addressing some of the common questions and concerns that readers may have.

For more information on evaluating and simplifying functions, check out the following resources:

• Khan Academy: Evaluating and Simplifying Functions
• Mathway: Evaluating and Simplifying Functions
• Wolfram Alpha: Evaluating and Simplifying Functions

Q: What is the value of $f(0)$? A: The value of $f(0)$ is $14$.

Q: Can the function $f(x) = 7\left(x^4 + 2\right)$ be used to model population growth? A: Yes, the function can be used to model population growth.

Q: How can I graph the function $f(x) = 7\left(x^4 + 2\right)$? A: You can use a coordinate plane and plot the points $(x, f(x))$ for various values of $x$.

Q: What is the domain of the function $f(x) = 7\left(x^4 + 2\right)$? A: The domain of the function is all real numbers.

Q: Can the function $f(x) = 7\left(x^4 + 2\right)$ be used to model economic growth? A: Yes, the function can be used to model economic growth.