Two Functions, { F $}$ And { G $} , A R E D E F I N E D A S F O L L O W S : , Are Defined As Follows: , A Re D E F In E D A S F O Ll O W S : $ F X \rightarrow 2x - 1 $ $ $ G X \rightarrow 4x - 2 $ $where { X $}$ Is A Real Number.a) If { F(x) - G(x) \leq 0 $}$, Find
Understanding the Functions f(x) and g(x)
In mathematics, functions are used to describe the relationship between two variables. In this case, we have two functions, f(x) and g(x), defined as follows:
- f(x) = 2x - 1
- g(x) = 4x - 2
where x is a real number. These functions can be used to model various real-world scenarios, such as population growth, financial transactions, or physical phenomena.
Analyzing the Difference Between f(x) and g(x)
To understand the relationship between f(x) and g(x), we need to find the difference between the two functions. This can be done by subtracting g(x) from f(x):
f(x) - g(x) = (2x - 1) - (4x - 2)
Simplifying the expression, we get:
f(x) - g(x) = 2x - 1 - 4x + 2
Combine like terms:
f(x) - g(x) = -2x + 1
Now, we need to find the values of x for which f(x) - g(x) ≤ 0.
Solving the Inequality f(x) - g(x) ≤ 0
To solve the inequality f(x) - g(x) ≤ 0, we need to find the values of x that make the expression -2x + 1 less than or equal to zero.
-2x + 1 ≤ 0
Subtract 1 from both sides:
-2x ≤ -1
Divide both sides by -2:
x ≥ 1/2
Therefore, the values of x for which f(x) - g(x) ≤ 0 are x ≥ 1/2.
Graphical Representation
To visualize the relationship between f(x) and g(x), we can plot the two functions on a coordinate plane. The graph of f(x) = 2x - 1 is a straight line with a slope of 2 and a y-intercept of -1. The graph of g(x) = 4x - 2 is also a straight line with a slope of 4 and a y-intercept of -2.
The graph of f(x) - g(x) = -2x + 1 is a straight line with a slope of -2 and a y-intercept of 1. The graph of f(x) - g(x) ≤ 0 is the region below the graph of f(x) - g(x) = -2x + 1.
Conclusion
In conclusion, we have analyzed the functions f(x) and g(x) and found the values of x for which f(x) - g(x) ≤ 0. We have also provided a graphical representation of the relationship between the two functions. This analysis can be used to model various real-world scenarios and make informed decisions.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Algebra, 2nd edition, Michael Artin
Further Reading
- For more information on functions and inequalities, please refer to the following resources:
- Calculus, 3rd edition, Michael Spivak
- Algebra, 2nd edition, Michael Artin
- Introduction to Mathematical Analysis, 2nd edition, Richard R. Goldberg
FAQs
- Q: What is the difference between f(x) and g(x)? A: The difference between f(x) and g(x) is -2x + 1.
- Q: What values of x make f(x) - g(x) ≤ 0?
A: The values of x that make f(x) - g(x) ≤ 0 are x ≥ 1/2.
Q&A: Functions f(x) and g(x)
Q: What are the functions f(x) and g(x)?
A: The functions f(x) and g(x) are defined as follows:
- f(x) = 2x - 1
- g(x) = 4x - 2
where x is a real number.
Q: What is the difference between f(x) and g(x)?
A: The difference between f(x) and g(x) is -2x + 1.
Q: How do I find the values of x for which f(x) - g(x) ≤ 0?
A: To find the values of x for which f(x) - g(x) ≤ 0, you need to solve the inequality -2x + 1 ≤ 0.
Q: What are the steps to solve the inequality -2x + 1 ≤ 0?
A: The steps to solve the inequality -2x + 1 ≤ 0 are:
- Subtract 1 from both sides: -2x ≤ -1
- Divide both sides by -2: x ≥ 1/2
Q: What is the graphical representation of the functions f(x) and g(x)?
A: The graphical representation of the functions f(x) and g(x) is a coordinate plane with the graph of f(x) = 2x - 1 as a straight line with a slope of 2 and a y-intercept of -1, and the graph of g(x) = 4x - 2 as a straight line with a slope of 4 and a y-intercept of -2.
Q: What is the graph of f(x) - g(x) ≤ 0?
A: The graph of f(x) - g(x) ≤ 0 is the region below the graph of f(x) - g(x) = -2x + 1.
Q: How can I use the functions f(x) and g(x) in real-world scenarios?
A: The functions f(x) and g(x) can be used to model various real-world scenarios, such as population growth, financial transactions, or physical phenomena.
Q: What are some common applications of the functions f(x) and g(x)?
A: Some common applications of the functions f(x) and g(x) include:
- Modeling population growth
- Analyzing financial transactions
- Studying physical phenomena
Q: What are some tips for working with the functions f(x) and g(x)?
A: Some tips for working with the functions f(x) and g(x) include:
- Make sure to understand the definitions of the functions
- Use algebraic manipulations to simplify expressions
- Graph the functions to visualize the relationships between them
Q: What are some common mistakes to avoid when working with the functions f(x) and g(x)?
A: Some common mistakes to avoid when working with the functions f(x) and g(x) include:
- Not understanding the definitions of the functions
- Not using algebraic manipulations to simplify expressions
- Not graphing the functions to visualize the relationships between them
Q: Where can I find more information on the functions f(x) and g(x)?
A: You can find more information on the functions f(x) and g(x) in the following resources:
- Calculus, 3rd edition, Michael Spivak
- Algebra, 2nd edition, Michael Artin
- Introduction to Mathematical Analysis, 2nd edition, Richard R. Goldberg