Understanding the Exclusion in the Function f(x) 5/x

Often, when dealing with mathematical functions, it is crucial to identify any values that are not in the domain of the function. This includes a type of function known as the excluded value. A prime example is the function f(x) 5/x. In this article, we will explore the concept of the excluded value for this particular function and provide a clear explanation of why some values are excluded.

What is the Excluded Value?

The excluded value of a function is the input value that makes the function undefined. In the context of f(x) 5/x, we need to determine what value of x would make the function undefined. This is important because any value that results in a division by zero is undefined in mathematics.

Why Division by Zero is Undefined

Division by zero is generally regarded as an undefined operation. The reason for this is that it contradicts the basic principles of mathematics. For example, if we were to divide 5 by 0, it would imply that there exists some number y such that 0 * y 5. However, no such number y exists, which is why the operation is undefined.

Identifying the Excluded Value

To find the excluded value for the function f(x) 5/x, we need to set the denominator equal to zero and solve for x: 5/x 0 x 0 Therefore, the excluded value is x 0. When x 0, the function f(x) 5/x becomes undefined because the denominator is zero.

Graphical Representation

The graph of the function f(x) 5/x is a hyperbola, which has a vertical asymptote at x 0. This vertical asymptote represents the point where the function is undefined due to the excluded value. The graph will show a sharp change in behavior as x gets closer to zero.

Constant Function vs. Linear Equations

It is important to distinguish between the function f(x) 5/x and a constant function. A constant function is defined as a function that always takes the same value regardless of the input. An example of a constant function is f(x) 5. This function is a horizontal line when graphed, indicating that the value of y (or f(x)) does not change as x changes.

Conclusion

Understanding the concept of the excluded value in functions is essential for correctly analyzing and graphing mathematical functions. For the function f(x) 5/x, the excluded value is x 0. This value is excluded because it results in an undefined operation. Recognizing and understanding such exclusions is crucial for a comprehensive knowledge of mathematical functions and their behavior.