The Functions F ( X ) = X 3 + X 2 − 2 X + 3 F(x)=x^3+x^2-2x+3 F ( X ) = X 3 + X 2 − 2 X + 3 And G ( X ) = Log ⁡ ( X ) + 2 G(x)=\log(x)+2 G ( X ) = Lo G ( X ) + 2 Are Given.Part A: What Type Of Functions Are F ( X F(x F ( X ] And G ( X G(x G ( X ]? Justify Your Answer.Part B: Find The Domain And Range For F ( X F(x F ( X ] And G ( X G(x G ( X ].

Mar 13, 2025 by ADMIN 347 views

$The Functions $f(x)=x^3+x^2-2x+3$ and $g(x)=\log(x)+2$: Analysis and Domain/Range$

Part A: Type of Functions

The given functions $f(x)=x^3+x^2-2x+3$ and $g(x)=\log(x)+2$ can be classified based on their characteristics. To determine the type of functions, we need to analyze their properties.

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

The function $f(x)=x^3+x^2-2x+3$ is a polynomial function. This is because it is defined as the sum of terms, where each term is a product of a variable and a coefficient. The highest power of the variable $x$ in this function is 3, which means it is a cubic polynomial.

Function $g(x)=\log(x)+2$

The function $g(x)=\log(x)+2$ is a logarithmic function. This is because it is defined as the logarithm of the variable $x$ with a base that is not explicitly mentioned. The logarithmic function is a fundamental function in mathematics that is used to describe the relationship between two quantities.

Justification

To justify the classification of these functions, we can consider the following properties:

Polynomial functions: These functions are defined as the sum of terms, where each term is a product of a variable and a coefficient. They can be classified based on the highest power of the variable, such as linear, quadratic, cubic, etc.
Logarithmic functions: These functions are defined as the logarithm of the variable with a base that is not explicitly mentioned. They are used to describe the relationship between two quantities.

Part B: Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values that the function can produce.

Domain of $f(x)=x^3+x^2-2x+3$

The domain of $f(x)=x^3+x^2-2x+3$ is all real numbers, denoted as $\mathbb{R}$ . This is because the function is defined for all real values of $x$ .

Range of $f(x)=x^3+x^2-2x+3$

The range of $f(x)=x^3+x^2-2x+3$ is also all real numbers, denoted as $\mathbb{R}$ . This is because the function can produce any real value as output.

Domain of $g(x)=\log(x)+2$

The domain of $g(x)=\log(x)+2$ is all positive real numbers, denoted as $(0, \infty)$ . This is because the logarithmic function is defined only for positive values of $x$ .

Range of $g(x)=\log(x)+2$

The range of $g(x)=\log(x)+2$ is all real numbers, denoted as $\mathbb{R}$ . This is because the function can produce any real value as output.

Conclusion

Discussion

The functions $f(x)$ and $g(x)$ have different properties and characteristics. The polynomial function $f(x)$ is defined for all real numbers, while the logarithmic function $g(x)$ is defined only for positive real numbers. The range of both functions is all real numbers, which means they can produce any real value as output.

Applications

The functions $f(x)$ and $g(x)$ have various applications in mathematics and other fields. The polynomial function $f(x)$ can be used to model real-world phenomena, such as population growth or chemical reactions. The logarithmic function $g(x)$ can be used to describe the relationship between two quantities, such as the relationship between the volume of a gas and its temperature.

Future Work

In the future, it would be interesting to explore the properties of these functions further. For example, we could investigate the behavior of the functions as $x$ approaches infinity or negative infinity. We could also explore the relationship between the functions and other mathematical concepts, such as calculus or differential equations.

References

[1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
[2] "Logarithmic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html

Keywords

Polynomial functions
Logarithmic functions
Domain
Range
Real numbers
Positive real numbers
Applications
Mathematics

Frequently Asked Questions

Q: What type of function is $f(x)=x^3+x^2-2x+3$ ?

A: The function $f(x)=x^3+x^2-2x+3$ is a polynomial function. This is because it is defined as the sum of terms, where each term is a product of a variable and a coefficient.

Q: What type of function is $g(x)=\log(x)+2$ ?

A: The function $g(x)=\log(x)+2$ is a logarithmic function. This is because it is defined as the logarithm of the variable $x$ with a base that is not explicitly mentioned.

Q: What is the domain of $f(x)=x^3+x^2-2x+3$ ?

A: The domain of $f(x)=x^3+x^2-2x+3$ is all real numbers, denoted as $\mathbb{R}$ . This is because the function is defined for all real values of $x$ .

Q: What is the range of $f(x)=x^3+x^2-2x+3$ ?

A: The range of $f(x)=x^3+x^2-2x+3$ is also all real numbers, denoted as $\mathbb{R}$ . This is because the function can produce any real value as output.

Q: What is the domain of $g(x)=\log(x)+2$ ?

A: The domain of $g(x)=\log(x)+2$ is all positive real numbers, denoted as $(0, \infty)$ . This is because the logarithmic function is defined only for positive values of $x$ .

Q: What is the range of $g(x)=\log(x)+2$ ?

A: The range of $g(x)=\log(x)+2$ is all real numbers, denoted as $\mathbb{R}$ . This is because the function can produce any real value as output.

Q: Can the functions $f(x)$ and $g(x)$ be used to model real-world phenomena?

A: Yes, the functions $f(x)$ and $g(x)$ can be used to model real-world phenomena. The polynomial function $f(x)$ can be used to model population growth or chemical reactions, while the logarithmic function $g(x)$ can be used to describe the relationship between two quantities.

Q: What are some applications of the functions $f(x)$ and $g(x)$ ?

A: Some applications of the functions $f(x)$ and $g(x)$ include:

Modeling population growth or chemical reactions using the polynomial function $f(x)$
Describing the relationship between two quantities using the logarithmic function $g(x)$
Analyzing the behavior of the functions as $x$ approaches infinity or negative infinity
Investigating the relationship between the functions and other mathematical concepts, such as calculus or differential equations

Q: What are some future directions for research on the functions $f(x)$ and $g(x)$ ?

A: Some future directions for research on the functions $f(x)$ and $g(x)$ include:

Investigating the behavior of the functions as $x$ approaches infinity or negative infinity
Analyzing the relationship between the functions and other mathematical concepts, such as calculus or differential equations
Developing new applications for the functions $f(x)$ and $g(x)$ in fields such as physics, engineering, or economics

Conclusion

In conclusion, the functions $f(x)=x^3+x^2-2x+3$ and $g(x)=\log(x)+2$ are a polynomial function and a logarithmic function, respectively. The domain of $f(x)$ is all real numbers, while the domain of $g(x)$ is all positive real numbers. The range of both functions is all real numbers. These functions have various applications in mathematics and other fields, and there are many future directions for research on these functions.

References

[1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
[2] "Logarithmic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html

Keywords

Polynomial functions
Logarithmic functions
Domain
Range
Real numbers
Positive real numbers
Applications
Mathematics

Part A: Type of Functions

Function f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3

Function g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2

Justification

Part B: Domain and Range

Domain of f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3

Range of f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3

Domain of g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2

Range of g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2

Conclusion

Discussion

Applications

Future Work

References

Keywords

Frequently Asked Questions

Q: What type of function is f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3?

Q: What type of function is g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2?

Q: What is the domain of f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3?

Q: What is the range of f(x)=x3+x2−2x+3f(x)=x^3+x^2-2x+3f(x)=x3+x2−2x+3?

Q: What is the domain of g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2?

Q: What is the range of g(x)=log⁡(x)+2g(x)=\log(x)+2g(x)=log(x)+2?

Q: Can the functions f(x)f(x)f(x) and g(x)g(x)g(x) be used to model real-world phenomena?

Q: What are some applications of the functions f(x)f(x)f(x) and g(x)g(x)g(x)?

Q: What are some future directions for research on the functions f(x)f(x)f(x) and g(x)g(x)g(x)?

Conclusion

References

Keywords

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

Function $g(x)=\log(x)+2$

Domain of $f(x)=x^3+x^2-2x+3$

Range of $f(x)=x^3+x^2-2x+3$

Domain of $g(x)=\log(x)+2$

Range of $g(x)=\log(x)+2$

Q: What type of function is $f(x)=x^3+x^2-2x+3$ ?

Q: What type of function is $g(x)=\log(x)+2$ ?

Q: What is the domain of $f(x)=x^3+x^2-2x+3$ ?

Q: What is the range of $f(x)=x^3+x^2-2x+3$ ?

Q: What is the domain of $g(x)=\log(x)+2$ ?

Q: What is the range of $g(x)=\log(x)+2$ ?

Q: Can the functions $f(x)$ and $g(x)$ be used to model real-world phenomena?

Q: What are some applications of the functions $f(x)$ and $g(x)$ ?

Q: What are some future directions for research on the functions $f(x)$ and $g(x)$ ?