Functions $f(x$\] And $g(x$\] Are Defined Below:$\begin{array}{l} f(x)=\frac{1}{x-3}+1 \\ g(x)=2 \sqrt{x-3} \end{array}$Determine Where $f(x)=g(x$\] By Graphing.A. $x=1$ B. $x=3$ C. $x=2$

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Introduction

In this article, we will explore the problem of determining where the functions $f(x)$ and $g(x)$ are equal by graphing. The functions $f(x)$ and $g(x)$ are defined as follows:

$$\begin{array}{l} f(x)=\frac{1}{x-3}+1 \\ g(x)=2 \sqrt{x-3} \end{array}$$

We will use graphing to find the solution to the equation $f(x) = g(x)$.

Graphing the Functions

To graph the functions $f(x)$ and $g(x)$, we can use a graphing calculator or a computer algebra system. We will first graph the function $f(x)$.

Graph of $f(x)$

The graph of $f(x)$ is a hyperbola with a vertical asymptote at $x = 3$. The graph is shown below:

Graph of $g(x)$

The graph of $g(x)$ is a parabola with a horizontal asymptote at $y = 0$. The graph is shown below:

Finding the Solution

To find the solution to the equation $f(x) = g(x)$, we need to find the point of intersection between the two graphs. We can do this by setting the two functions equal to each other and solving for $x$.

$$\frac{1}{x-3}+1 = 2 \sqrt{x-3}$$

We can simplify the equation by multiplying both sides by $x-3$.

$$1 + x - 3 = 2 \sqrt{(x-3)^2}$$

We can further simplify the equation by squaring both sides.

$$4 = 4(x-3)$$

We can solve for $x$ by dividing both sides by $4$.

$$1 = x - 3$$

We can add $3$ to both sides to get the final solution.

$$x = 4$$

However, we need to check if this solution is valid. We can do this by plugging $x = 4$ into both functions and checking if they are equal.

$$f(4) = \frac{1}{4-3}+1 = 2$$

$$g(4) = 2 \sqrt{4-3} = 2$$

Since $f(4) = g(4)$, we have found the solution to the equation $f(x) = g(x)$.

Conclusion

In this article, we used graphing to find the solution to the equation $f(x) = g(x)$. We graphed the functions $f(x)$ and $g(x)$ and found the point of intersection between the two graphs. We then solved for $x$ and checked if the solution was valid. We found that the solution to the equation $f(x) = g(x)$ is $x = 4$.

Discussion

The problem of finding the solution to the equation $f(x) = g(x)$ is a classic problem in mathematics. It requires the use of graphing and algebraic techniques to find the solution. The solution to the equation $f(x) = g(x)$ is $x = 4$, which is a valid solution.

References

• [1] "Graphing Functions" by Math Open Reference
• [2] "Algebraic Techniques" by Khan Academy

Keywords

• Graphing functions
• Algebraic techniques
• Equation solving
• Mathematics

Category

• Mathematics

Tags

• Graphing functions
• Algebraic techniques
• Equation solving
• Mathematics

Related Articles

• "Solving Equations by Graphing"
• "Graphing Functions with a Graphing Calculator"
• "Algebraic Techniques for Solving Equations"
  Q&A: Solving the Equation $f(x) = g(x)$ by Graphing =====================================================

Introduction

In our previous article, we explored the problem of determining where the functions $f(x)$ and $g(x)$ are equal by graphing. We graphed the functions $f(x)$ and $g(x)$ and found the point of intersection between the two graphs. In this article, we will answer some frequently asked questions about solving the equation $f(x) = g(x)$ by graphing.

Q: What is the equation $f(x) = g(x)$?

A: The equation $f(x) = g(x)$ is a mathematical equation that states that the function $f(x)$ is equal to the function $g(x)$.

Q: How do I graph the functions $f(x)$ and $g(x)$?

A: To graph the functions $f(x)$ and $g(x)$, you can use a graphing calculator or a computer algebra system. You can also use a graphing app or a online graphing tool.

Q: What is the point of intersection between the two graphs?

A: The point of intersection between the two graphs is the point where the two functions are equal. In this case, the point of intersection is $x = 4$.

Q: How do I find the solution to the equation $f(x) = g(x)$?

A: To find the solution to the equation $f(x) = g(x)$, you need to find the point of intersection between the two graphs. You can do this by graphing the functions $f(x)$ and $g(x)$ and finding the point where they intersect.

Q: What are some common mistakes to avoid when solving the equation $f(x) = g(x)$?

A: Some common mistakes to avoid when solving the equation $f(x) = g(x)$ include:

• Not graphing the functions $f(x)$ and $g(x)$ correctly
• Not finding the point of intersection between the two graphs
• Not checking if the solution is valid

Q: What are some real-world applications of solving the equation $f(x) = g(x)$?

A: Some real-world applications of solving the equation $f(x) = g(x)$ include:

• Finding the point of intersection between two curves in a graph
• Solving systems of equations
• Finding the maximum or minimum value of a function

Q: How can I practice solving the equation $f(x) = g(x)$?

A: You can practice solving the equation $f(x) = g(x)$ by graphing the functions $f(x)$ and $g(x)$ and finding the point of intersection between the two graphs. You can also try solving different types of equations, such as linear and quadratic equations.

Conclusion

In this article, we answered some frequently asked questions about solving the equation $f(x) = g(x)$ by graphing. We covered topics such as graphing the functions $f(x)$ and $g(x)$, finding the point of intersection between the two graphs, and common mistakes to avoid. We also discussed some real-world applications of solving the equation $f(x) = g(x)$ and provided some tips for practicing solving the equation.

Discussion

The equation $f(x) = g(x)$ is a fundamental concept in mathematics that has many real-world applications. Solving the equation $f(x) = g(x)$ requires graphing the functions $f(x)$ and $g(x)$ and finding the point of intersection between the two graphs. By practicing solving the equation $f(x) = g(x)$, you can improve your graphing skills and develop a deeper understanding of mathematical concepts.

References

• [1] "Graphing Functions" by Math Open Reference
• [2] "Algebraic Techniques" by Khan Academy

Keywords

• Graphing functions
• Algebraic techniques
• Equation solving
• Mathematics

Category

• Mathematics

Tags

• Graphing functions
• Algebraic techniques
• Equation solving
• Mathematics

Related Articles

• "Solving Equations by Graphing"
• "Graphing Functions with a Graphing Calculator"
• "Algebraic Techniques for Solving Equations"