How To Find The X-intercept Of F(x) = 64/(x-2) + 4

To determine the x-intercept of the function $f(x) = \frac{64}{x-2} + 4$, we need to find the value(s) of x for which $f(x) = 0$. The x-intercept is the point where the graph of the function crosses the x-axis, and at this point, the y-coordinate (or the function's value) is zero. This involves algebraic manipulation and solving for x. The process includes setting the function equal to zero, isolating the fractional term, and then solving the resulting equation. Understanding how to find x-intercepts is fundamental in analyzing functions and their graphs, as these points provide critical information about the function's behavior and its relationship to the coordinate axes.

Setting Up the Equation

The first step in finding the x-intercept is to set the function $f(x)$ equal to zero. This gives us the equation:

$\frac{64}{x-2} + 4 = 0$

This equation represents the condition we need to satisfy to find the x-intercept. The x-intercept is the value of x where the function's output is zero, graphically represented as the point where the function's graph intersects the x-axis. Setting the function to zero allows us to find these specific x values. This process is essential in various mathematical and real-world applications, providing key insights into the behavior of the function. For instance, in physics, it might represent the point where a projectile hits the ground, or in economics, it might signify the break-even point where costs equal revenue.

Isolating the Fractional Term

Next, we need to isolate the fractional term $\frac{64}{x-2}$ on one side of the equation. To do this, we subtract 4 from both sides of the equation:

$\frac{64}{x-2} = -4$

Isolating the fractional term is a crucial step in solving the equation. This manipulation allows us to work more directly with the term containing x in the denominator. By getting the fractional term alone, we set the stage for further algebraic steps, such as cross-multiplication or finding a common denominator. This is a common strategy in solving equations involving fractions and is applicable in many different mathematical contexts. The goal here is to simplify the equation into a form that is easier to solve, and isolating the fractional term is a key part of that simplification process.

Solving for x

To solve for x, we can multiply both sides of the equation by $(x-2)$ to eliminate the denominator:

$64 = -4(x-2)$

This step eliminates the fraction, making the equation easier to solve. Now, distribute the -4 on the right side:

$64 = -4x + 8$

Next, subtract 8 from both sides:

$56 = -4x$

Finally, divide both sides by -4:

$x = -14$

This gives us the x-coordinate of the x-intercept. The process of solving for x involves several algebraic manipulations, including multiplication, distribution, subtraction, and division. Each step is performed to isolate x and find its value, which is the x-coordinate of the x-intercept. This method is widely used in algebra and calculus to find the roots or zeros of functions. The solution $x = -14$ indicates where the function intersects the x-axis, providing a critical point for understanding the function's behavior and graph.

The X-intercept

Therefore, the x-intercept of the function $f(x) = \frac{64}{x-2} + 4$ is $(-14, 0)$. The x-intercept is the point where the graph of the function crosses the x-axis, and it is represented as a coordinate pair where the y-coordinate is zero. In this case, we found that when $x = -14$, the function $f(x)$ equals zero, indicating that the graph intersects the x-axis at the point $(-14, 0)$. Understanding the x-intercept is important for graphing the function and for interpreting the function's behavior. It is a key feature in the analysis of functions, providing a concrete point of reference on the coordinate plane.

Verification

To verify our solution, we can plug $x = -14$ back into the original function:

$f(-14) = \frac{64}{-14-2} + 4$

$f(-14) = \frac{64}{-16} + 4$

$f(-14) = -4 + 4$

$f(-14) = 0$

Since $f(-14) = 0$, our solution is correct. Verification is a crucial step in problem-solving, as it ensures that the calculated solution satisfies the original equation or condition. By plugging the value back into the function, we can confirm that it indeed results in the expected outcome. In this case, plugging $x = -14$ into the function yields $f(-14) = 0$, which confirms that $(-14, 0)$ is indeed the x-intercept of the function. This step reinforces the accuracy of our algebraic manipulations and solution.

In conclusion, the x-intercept of the function $f(x) = \frac{64}{x-2} + 4$ is $(-14, 0)$. This is found by setting the function equal to zero, isolating the fractional term, solving for x, and then verifying the solution. Understanding how to find x-intercepts is a fundamental skill in mathematics and is essential for analyzing and graphing functions. The process involves several algebraic techniques and provides a key point of reference for understanding the behavior of the function on the coordinate plane. This skill is applicable in various contexts, from academic problem-solving to real-world applications in fields such as physics, engineering, and economics.