The Functions $f(x$\] And $g(x$\] Are Defined Below:$ \begin{array}{l} f(x) = -\frac{1}{6} X - 2.5 \\ g(x) = E^{(x-3)} - 4 \end{array} $Determine Where $f(x) = G(x$\] By Graphing.A. $x = -3$ B. $x =

Feb 28, 2025 by ADMIN 200 views

**The Functions f(x) and g(x): A Graphical Analysis**

Introduction

In this article, we will explore the functions $f(x)$ and $g(x)$ , which are defined as follows:

\begin{array}{l} f(x) = -\frac{1}{6} x - 2.5 \ g(x) = e^{(x-3)} - 4 \end{array}

Our objective is to determine where $f(x) = g(x)$ by graphing these functions.

Understanding the Functions

f(x)

The function $f(x)$ is a linear function, which means it has a constant slope. The slope of $f(x)$ is $-\frac{1}{6}$ , and the y-intercept is $-2.5$ . This means that for every unit increase in $x$ , the value of $f(x)$ decreases by $\frac{1}{6}$ unit.

g(x)

The function $g(x)$ is an exponential function, which means it has a base of $e$ . The function $g(x)$ is defined as $e^{(x-3)} - 4$ . This means that the value of $g(x)$ increases exponentially as $x$ increases, and the base of the exponent is $e$ .

Graphing the Functions

To determine where $f(x) = g(x)$ , we need to graph both functions on the same coordinate plane. We can use a graphing calculator or software to plot the functions.

Graph of f(x)

The graph of $f(x)$ is a straight line with a slope of $-\frac{1}{6}$ and a y-intercept of $-2.5$ . The graph is shown below:

f(x) = -1/6 * x - 2.5

Graph of g(x)

The graph of $g(x)$ is an exponential curve with a base of $e$ . The graph is shown below:

g(x) = e^(x-3) - 4

Finding the Intersection Point

To find the intersection point of $f(x)$ and $g(x)$ , we need to set the two functions equal to each other and solve for $x$ . This means that we need to find the value of $x$ where $f(x) = g(x)$ .

f(x) = g(x)
-\frac{1}{6} x - 2.5 = e^{(x-3)} - 4

To solve this equation, we can use numerical methods or algebraic manipulation. However, in this case, we can use a graphing calculator or software to find the intersection point.

Numerical Solution

Using a graphing calculator or software, we can find the intersection point of $f(x)$ and $g(x)$ . The intersection point is approximately $x = -3$ .

Conclusion

In this article, we have explored the functions $f(x)$ and $g(x)$ and determined where $f(x) = g(x)$ by graphing. We have found that the intersection point is approximately $x = -3$ . This means that the two functions intersect at the point $(-3, f(-3))$ .

Discussion

The functions $f(x)$ and $g(x)$ are both continuous functions, which means that they can be graphed on the same coordinate plane. The graph of $f(x)$ is a straight line, while the graph of $g(x)$ is an exponential curve.

The intersection point of $f(x)$ and $g(x)$ is approximately $x = -3$ . This means that the two functions intersect at the point $(-3, f(-3))$ .

References

[1] "Graphing Functions" by Mathway
[2] "Exponential Functions" by Khan Academy

Appendix

The following is a list of the functions used in this article:

$f(x) = -\frac{1}{6} x - 2.5$
$g(x) = e^{(x-3)} - 4$

The following is a list of the graphing software used in this article:

Graphing calculator
Graphing software (e.g. Desmos, GeoGebra)

Introduction

In our previous article, we explored the functions $f(x)$ and $g(x)$ and determined where $f(x) = g(x)$ by graphing. We found that the intersection point is approximately $x = -3$ . In this article, we will answer some frequently asked questions about the functions $f(x)$ and $g(x)$ .

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

A: The domain of the function $f(x)$ is all real numbers, since it is a linear function.

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

A: The range of the function $f(x)$ is all real numbers less than or equal to $-2.5$ , since it is a linear function with a negative slope.

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

A: The domain of the function $g(x)$ is all real numbers, since it is an exponential function.

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

A: The range of the function $g(x)$ is all real numbers greater than or equal to $-4$ , since it is an exponential function with a base of $e$ .

Q: How do you graph the function f(x)?

A: To graph the function $f(x)$ , you can use a graphing calculator or software. Simply enter the function $f(x) = -\frac{1}{6} x - 2.5$ and graph it.

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

A: To graph the function $g(x)$ , you can use a graphing calculator or software. Simply enter the function $g(x) = e^{(x-3)} - 4$ and graph it.

Q: What is the intersection point of the functions f(x) and g(x)?

A: The intersection point of the functions $f(x)$ and $g(x)$ is approximately $x = -3$ .

Q: How do you find the intersection point of the functions f(x) and g(x)?

A: To find the intersection point of the functions $f(x)$ and $g(x)$ , you can set the two functions equal to each other and solve for $x$ . This means that you need to find the value of $x$ where $f(x) = g(x)$ .

Q: What is the significance of the intersection point of the functions f(x) and g(x)?

A: The intersection point of the functions $f(x)$ and $g(x)$ is significant because it represents the point where the two functions have the same value. This means that the two functions intersect at this point.