Question

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x}{x^{2}+x+2} $$

Short answer

The function is continuous on the interval \((-\infty, +\infty)\)

Step-by-step solution

Analyze the given function

The function given is \( f(x)=\frac{x}{x^{2}+x+2} \). In order for the function to be continuous, it needs to be defined for all x in its domain.

Determine where the function is undefined

To find where this function is undefined, look for values of x that would make the denominator zero. Solve the equation \(x^{2}+x+2=0\). However, if you tries to solve this quadratic equation, you'll see that it has no real solution, which means that the denominator never becomes zero.

Conclusion

Since there is no real x that makes the denominator zero, the function is continuous for all real x. Therefore, the interval of continuity is \((-\infty, +\infty)\).

Key concepts

Interval of Continuity

Continuity refers to a function being smooth without any breaks, jumps, or holes in its graph. To say a function is continuous on an interval means that it can be drawn without lifting your pencil from start to end of that interval. For the given function \( f(x)=\frac{x}{x^2+x+2} \), the task is to establish the intervals over which it is continuous.
First, identify potential points of discontinuity. This primarily happens where the function might be undefined. If a function remains defined across its entire domain, it implies continuity. Given that the denominator \( x^2 + x + 2 \) has no real roots, the fraction remains valid for all real \( x \).

Thus, there are no breaks in this function, making it continuous over the entire set of real numbers. The **interval of continuity** for this function is \( (-\infty, \infty) \). Continuous functions like these are smooth and unbroken, which simplifies both theoretical and applied mathematics.

Undefined Function

Function undefined points are crucial in identifying discontinuity. A function could be undefined if its denominator equals zero, since division by zero leads to undefined values. For equations like \( f(x)=\frac{x}{x^2+x+2} \), setting the denominator equal to zero helps locate these values.
Solving \( x^2 + x + 2 = 0 \) might seem a usual step, but here, discerning that it has no real solutions is important. This indicates the denominator never reaches zero for any real number, which means the function is not undefined at any point.

In simpler terms, if you can't find \( x \) values that make a denominator zero, the function remains defined across its domain. For students, understanding where a function is undefined helps in completing a picture of its continuity, thereby predicting its behavior more accurately.

Domain of a Function

The domain of a function encompasses all the possible values that \( x \) can take, such that the function gives a valid output. It's important for understanding where a function is valid and operative. For the function \( f(x)=\frac{x}{x^2+x+2} \), the domain is all real \( x \) except where the denominator becomes zero.
We solved to find \( x^2 + x + 2 = 0 \) does not yield any real roots. That confirms the denominator does not become zero, assuring us the function stays valid no matter what real number \( x \) is chosen. Thus, its domain is the entire set of real numbers, expressed as \( (-\infty, \infty) \).

Understanding the domain is key in both simple function evaluations and complex calculus solutions, as it ensures the inputs produce meaningful and defined outputs. Hence, recognizing and correctly determining a function's domain is an integral part of mastering calculus.