A Function \[$ G(x) \$\] Is Defined As Shown:$\[ g(x)=\left\{ \begin{array}{ll} 2+3x, & 0 \leq X\ \textless \ 4 \\ 0.5x+10, & 4 \leq X\ \textless \ 8 \\ 16, & X \geq 8 \end{array} \right. \\]What Is The Value Of \[$ G(4)

Feb 27, 2025 by ADMIN 224 views

A Comprehensive Analysis of the Function g(x)

In mathematics, functions are used to describe the relationship between variables. A function g(x) is defined as a set of rules that take an input value x and produce an output value g(x). In this article, we will analyze a specific function g(x) and determine its value for a given input.

The function g(x) is defined as:

${ g(x)=\left\{ \begin{array}{ll} 2+3x, & 0 \leq x\ \textless \ 4 \\ 0.5x+10, & 4 \leq x\ \textless \ 8 \\ 16, & x \geq 8 \end{array} \right. \}$

This function has three different rules, each corresponding to a specific range of input values. The first rule applies when 0 ≤ x < 4, the second rule applies when 4 ≤ x < 8, and the third rule applies when x ≥ 8.

To determine the value of g(4), we need to analyze the function and identify the rule that applies to the input x = 4.

For the first rule, 0 ≤ x < 4, we can see that x = 4 is not within this range. Therefore, this rule does not apply.
For the second rule, 4 ≤ x < 8, we can see that x = 4 is within this range. Therefore, this rule applies.
For the third rule, x ≥ 8, we can see that x = 4 is not within this range. Therefore, this rule does not apply.

Since the second rule applies to the input x = 4, we can use this rule to determine the value of g(4).

The second rule is given by:

g(x) = 0.5x + 10

Substituting x = 4 into this equation, we get:

g(4) = 0.5(4) + 10 g(4) = 2 + 10 g(4) = 12

Therefore, the value of g(4) is 12.

In this article, we analyzed a specific function g(x) and determined its value for a given input. We identified the rule that applies to the input x = 4 and used this rule to calculate the value of g(4). The value of g(4) is 12.

Functions are used to describe the relationship between variables.
A function g(x) is defined as a set of rules that take an input value x and produce an output value g(x).
The function g(x) has three different rules, each corresponding to a specific range of input values.
To determine the value of g(4), we need to analyze the function and identify the rule that applies to the input x = 4.
The value of g(4) is 12.

In future articles, we can explore more complex functions and analyze their behavior. We can also investigate the applications of functions in various fields, such as physics, engineering, and economics.

[1] "Functions" by Khan Academy
[2] "Algebra" by MIT OpenCourseWare

Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
Rule: A specific equation or formula that defines the function for a given range of input values.
Input: A value that is fed into the function.
Output: The value that is produced by the function for a given input.
A Comprehensive Q&A on the Function g(x)

In our previous article, we analyzed a specific function g(x) and determined its value for a given input. In this article, we will answer some frequently asked questions about the function g(x) and provide additional insights into its behavior.

Q: What is the domain of the function g(x)?

A: The domain of the function g(x) is the set of all possible input values x. In this case, the domain is divided into three parts:

For the first rule, 0 ≤ x < 4, the domain is [0, 4).
For the second rule, 4 ≤ x < 8, the domain is [4, 8).
For the third rule, x ≥ 8, the domain is [8, ∞).

Q: What is the range of the function g(x)?

A: The range of the function g(x) is the set of all possible output values g(x). In this case, the range is:

For the first rule, 2 + 3x, the range is (2, ∞).
For the second rule, 0.5x + 10, the range is [10, ∞).
For the third rule, 16, the range is {16}.

Q: How do I determine which rule applies to a given input x?

A: To determine which rule applies to a given input x, you need to check which range x falls into. For example, if x = 5, then x falls into the range [4, 8), so the second rule applies.

Q: Can I use the function g(x) to model real-world phenomena?

A: Yes, the function g(x) can be used to model real-world phenomena. For example, the function could represent the growth of a population over time, where the input x represents time and the output g(x) represents the population size.

Q: How do I graph the function g(x)?

A: To graph the function g(x), you need to plot the three rules on a coordinate plane. The first rule is a linear function with a slope of 3 and a y-intercept of 2. The second rule is a linear function with a slope of 0.5 and a y-intercept of 10. The third rule is a horizontal line at y = 16.

Q: Can I use the function g(x) to solve equations?

A: Yes, the function g(x) can be used to solve equations. For example, if we want to find the value of x that satisfies the equation g(x) = 12, we can use the second rule to solve for x.

Q: How do I find the inverse of the function g(x)?

A: To find the inverse of the function g(x), you need to swap the x and y variables and solve for y. This will give you the inverse function g^(-1)(x).

In this article, we answered some frequently asked questions about the function g(x) and provided additional insights into its behavior. We hope that this article has been helpful in understanding the function g(x) and its applications.

The domain of the function g(x) is divided into three parts: [0, 4), [4, 8), and [8, ∞).
The range of the function g(x) is divided into three parts: (2, ∞), [10, ∞), and {16}.
To determine which rule applies to a given input x, you need to check which range x falls into.
The function g(x) can be used to model real-world phenomena, such as population growth.
The function g(x) can be graphed by plotting the three rules on a coordinate plane.
The function g(x) can be used to solve equations and find the inverse of the function.

[1] "Functions" by Khan Academy
[2] "Algebra" by MIT OpenCourseWare

Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
Rule: A specific equation or formula that defines the function for a given range of input values.
Input: A value that is fed into the function.
Output: The value that is produced by the function for a given input.
Domain: The set of all possible input values x.
Range: The set of all possible output values g(x).
Inverse: A function that undoes the action of the original function.