Two Functions, { F $}$ And { G $} , A R E D E F I N E D B Y : , Are Defined By: , A Re D E F In E D B Y : { F(x) = 2x - 1 \} ${ G(x) = 4x - 2 }$where { X $}$ Is A Real Number.a) If { F(x) \leq 0 $}$, Find The Value Of { X $}$.b)

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Introduction

In mathematics, functions are used to describe the relationship between variables. In this article, we will explore two functions, { f $}$ and { g $}$, defined by the equations f(x)=2x−1{ f(x) = 2x - 1 } and g(x)=4x−2{ g(x) = 4x - 2 }, where { x $}$ is a real number. We will focus on solving the inequality { f(x) \leq 0 $}$ and find the value of { x $}$.

Understanding the Functions

Before we dive into solving the inequality, let's take a closer look at the two functions.

Function { f $}$

The function { f $}$ is defined by the equation f(x)=2x−1{ f(x) = 2x - 1 }. This is a linear function, which means it has a constant slope. The slope of the function is 2, and the y-intercept is -1.

Function { g $}$

The function { g $}$ is defined by the equation g(x)=4x−2{ g(x) = 4x - 2 }. This is also a linear function, with a slope of 4 and a y-intercept of -2.

Solving the Inequality

Now that we have a good understanding of the functions, let's focus on solving the inequality { f(x) \leq 0 $}$.

Step 1: Set Up the Inequality

To solve the inequality, we need to set up the equation f(x)≤0{ f(x) \leq 0 }. Substituting the definition of { f $}$, we get:

2x−1≤0{ 2x - 1 \leq 0 }

Step 2: Add 1 to Both Sides

To isolate the term with the variable, we need to add 1 to both sides of the inequality:

2x≤1{ 2x \leq 1 }

Step 3: Divide Both Sides by 2

To solve for { x $}$, we need to divide both sides of the inequality by 2:

x≤12{ x \leq \frac{1}{2} }

Conclusion

Therefore, the value of { x $}$ that satisfies the inequality { f(x) \leq 0 $}$ is { x \leq \frac{1}{2} $}$.

Graphical Representation

To visualize the solution, we can graph the function { f $}$ and shade the region where { f(x) \leq 0 $}$.

Graph of { f $}$

The graph of { f $}$ is a straight line with a slope of 2 and a y-intercept of -1.

Shaded Region

The shaded region represents the values of { x $}$ where { f(x) \leq 0 $}$. This region is to the left of the vertical line { x = \frac{1}{2} $}$.

Discussion

In this article, we have explored two functions, { f $}$ and { g $}$, defined by the equations f(x)=2x−1{ f(x) = 2x - 1 } and g(x)=4x−2{ g(x) = 4x - 2 }, where { x $}$ is a real number. We have focused on solving the inequality { f(x) \leq 0 $}$ and found the value of { x $}$. The solution is { x \leq \frac{1}{2} $}$.

Conclusion

In conclusion, solving inequalities with two functions requires a step-by-step approach. By understanding the functions and setting up the inequality, we can isolate the term with the variable and solve for { x $}$. The graphical representation of the solution provides a visual representation of the values of { x $}$ that satisfy the inequality.

References

  • [1] "Functions" by Khan Academy
  • [2] "Inequalities" by Math Open Reference

Further Reading

  • "Linear Functions" by Math Is Fun
  • "Inequalities with Two Variables" by Purplemath

FAQs

  • Q: What is the definition of the function { f $}$? A: The function { f $}$ is defined by the equation f(x)=2x−1{ f(x) = 2x - 1 }.
  • Q: What is the value of { x $}$ that satisfies the inequality { f(x) \leq 0 $}$? A: The value of { x $}$ that satisfies the inequality { f(x) \leq 0 $}$ is { x \leq \frac{1}{2} $}$.
    Frequently Asked Questions (FAQs) =====================================

Q: What is the definition of the function { f $}$?

A: The function { f $}$ is defined by the equation f(x)=2x−1{ f(x) = 2x - 1 }. This is a linear function, which means it has a constant slope. The slope of the function is 2, and the y-intercept is -1.

Q: What is the definition of the function { g $}$?

A: The function { g $}$ is defined by the equation g(x)=4x−2{ g(x) = 4x - 2 }. This is also a linear function, with a slope of 4 and a y-intercept of -2.

Q: How do I solve the inequality { f(x) \leq 0 $}$?

A: To solve the inequality { f(x) \leq 0 $}$, you need to follow these steps:

  1. Set up the equation f(x)≤0{ f(x) \leq 0 }.
  2. Add 1 to both sides of the equation.
  3. Divide both sides of the equation by 2.

Q: What is the value of { x $}$ that satisfies the inequality { f(x) \leq 0 $}$?

A: The value of { x $}$ that satisfies the inequality { f(x) \leq 0 $}$ is { x \leq \frac{1}{2} $}$.

Q: Can I use the same steps to solve the inequality { g(x) \leq 0 $}$?

A: No, you cannot use the same steps to solve the inequality { g(x) \leq 0 $}$. The function { g $}$ has a different slope and y-intercept than the function { f $}$, so you need to use a different approach to solve the inequality.

Q: How do I graph the function { f $}$ and shade the region where { f(x) \leq 0 $}$?

A: To graph the function { f $}$ and shade the region where { f(x) \leq 0 $}$, you need to follow these steps:

  1. Plot the graph of the function { f $}$.
  2. Identify the x-intercept of the function.
  3. Draw a vertical line at the x-intercept.
  4. Shade the region to the left of the vertical line.

Q: Can I use a calculator to solve the inequality { f(x) \leq 0 $}$?

A: Yes, you can use a calculator to solve the inequality { f(x) \leq 0 $}$. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: What is the difference between the functions { f $}$ and { g $}$?

A: The functions { f $}$ and { g $}$ have different slopes and y-intercepts. The function { f $}$ has a slope of 2 and a y-intercept of -1, while the function { g $}$ has a slope of 4 and a y-intercept of -2.

Q: Can I use the functions { f $}$ and { g $}$ to model real-world situations?

A: Yes, you can use the functions { f $}$ and { g $}$ to model real-world situations. For example, you can use the function { f $}$ to model the cost of producing a product, and the function { g $}$ to model the revenue generated by the product.

Q: What are some common applications of the functions { f $}$ and { g $}$?

A: Some common applications of the functions { f $}$ and { g $}$ include:

  • Modeling the cost of producing a product
  • Modeling the revenue generated by a product
  • Modeling the growth of a population
  • Modeling the spread of a disease

Q: Can I use the functions { f $}$ and { g $}$ to solve systems of equations?

A: Yes, you can use the functions { f $}$ and { g $}$ to solve systems of equations. However, you need to make sure that the functions are consistent with the system of equations.

Q: What are some common mistakes to avoid when working with the functions { f $}$ and { g $}$?

A: Some common mistakes to avoid when working with the functions { f $}$ and { g $}$ include:

  • Not checking the domain of the functions
  • Not checking the range of the functions
  • Not using the correct function to model a real-world situation
  • Not checking the consistency of the functions with a system of equations