Question

Use the equations \(y=\frac{1}{x}\) and \(y=\frac{1}{x+3}\). Explain why each equation is not a function for all real values of \(x .\)

Short answer

The equations \(y=\frac{1}{x}\) and \(y=\frac{1}{x+3}\) are not functions for all real values of \(x\) because both equations have specific values of \(x\) (0 and -3, respectively) for which they don't produce a valid output due to the undefined result of division by zero.

Step-by-step solution

Analyze the equation \(y=\frac{1}{x}\)

Here, notice that any value of \(x\) that makes the denominator zero would be undefined, as division by zero is undefined in mathematics. Thus, \(x\) cannot be zero.

Analyze the equation \(y=\frac{1}{x+3}\)

Similar to Step 1, \(x\) cannot be a value that makes the denominator zero. Therefore, when \(x+3=0\), \(x\) is equal to -3, so \(x\) cannot be -3. This makes this equation also not a function for all real values of \(x\).

Conclusion

Because both equations have restrictions on the possible values of \(x\), neither \(y=\frac{1}{x}\) nor \(y=\frac{1}{x+3}\) is a function for all real values of \(x\). Both equations exclude a specific value for \(x\) (0 and -3, respectively). Thus, they are not functions for every possible real number.

Key concepts

Domain of a Function

Understanding the domain of a function is akin to knowing the boundaries within which a function operates correctly. It comprises all the possible input values (commonly represented as 'x' values) that a function can accept without leading to any kind of mathematical error or contradiction.

For example, considering the equation \(y=\frac{1}{x}\), the domain excludes 0 because dividing by 0 is not defined and would disrupt the function's integrity. Essentially, the domain here would be all real numbers except 0. When we look at \(y=\frac{1}{x+3}\), the function once again omits a particular value from its domain, which is -3, to prevent division by zero. This means that each of these functions has its domain, which does NOT include the entire set of real numbers. Identifying a function's domain is crucial because it dictates how a function behaves and ensures that all operations within the function are valid.

Division by Zero

Division by zero is a concept that often perplexes students, but it's a fundamental rule in mathematics that division by zero is undefined. This fundamental rule impacts how we understand and work with functions in algebra.

No matter what value you attempt to divide by zero, the operation does not produce a meaningful result because a zero denominator signifies an attempt to divide something into zero parts—a puzzling proposition. In the context of functions like \(y=\frac{1}{x}\) and \(y=\frac{1}{x+3}\), division by zero is the cul-de-sac that sets the domain's limitations. If we overlook these limitations and attempt to plug zero into the denominators, we'd face an undefined expression, which is not permissible in the realm of functions. Thus, when dealing with algebraic functions, we must always keep an eye out to prevent division by zero to maintain meaningful and valid equations.

Undefined Values in Functions

Functions can occasionally lead us to undefined values, which are like potholes on the road of algebra—they must be circumvented to continue smoothly. These undefined values arise when the function's formula generates a mathematically impossible situation, such as division by zero or the square root of a negative number in real number calculations.

When resolving the equations \(y=\frac{1}{x}\) and \(y=\frac{1}{x+3}\), the undefined values occur at the input of 0 and -3, respectively. It's crucial to recognize these instances to avoid the misconception that these functions work for any and every real number. In practice, when we graph these functions or apply them to real-world problems, we must exclude these undefined values to maintain the functions' integrity and ensure they remain well-defined throughout their application.