Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=3-2 x-x^{2}\)

Short answer

The function \(f(x)=3-2x-x^{2}\) is continuous for all real numbers, \(-\infty<x<\infty\). There are no discontinuities, and hence no conditions of continuity are violated.

Step-by-step solution

Identify the function type

The given function \(f(x)=3-2x-x^{2}\) is a quadratic function. A polynomial function of degree n (in this case n=2, a quadratic function), is continuous everywhere in its domain, because all of the operations in the function (addition, subtraction, multiplication, and powers) are also continuous.

Determine the continuity

For a polynomial function, such as \(f(x) = 3-2x-x^{2}\), it is continuous for all values of real number x. The domain of \(f(x)\) is the set of all real numbers, \(-\infty<x<\infty\), and for every point in the domain, the function's limit exists and equals to its value. There are no discontinuities.

Identify the conditions not satisfied

Since \(f(x)=3-2x-x^{2}\) is continuous for all real numbers, no conditions of continuity are violated. Hence, this step does not apply because there's no discontinuity.

Key concepts

Quadratic Functions

When we talk about quadratic functions, we're discussing a very particular kind of polynomial. They're shaped like a U or an upside-down U and have a formula that can be written as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. The graph of such a function is called a parabola.

One remarkable property of quadratic functions is their symmetry; they are symmetric about a vertical line that goes through their vertex, the highest or lowest point on the graph. The importance of this symmetry becomes clear when we discuss the function's continuity and domain. Owing to the unbroken and smooth nature of the parabola, quadratic functions are inherently continuous throughout their domain.

Domain of a Function

The domain of a function is the set of all the inputs over which the function is defined. In simpler terms, it's an answer to the question, 'For which values of \(x\) does this function give a valid output?' For quadratic functions, like the one we're examining \(f(x) = 3-2x-x^2\), the domain is all real numbers because you can plug any real number into that function and get a real number out.

Understanding the domain is vital when studying the function's behavior because it tells us where to look for continuity and possible discontinuities. However, since quadratic functions are polynomial functions of degree 2, their continuous nature means we expect no interruptions in the graph — so the function is continuous over its entire domain.

Conditions of Continuity

For a function to be continuous at a point, it must satisfy three conditions: the function must be defined at that point, the limit of the function as it approaches the point from both sides must exist, and the value of the limit must equal the function's value at that point. These conditions ensure there are no breaks, jumps, or holes at that point in the graph.

For our quadratic function \(f(x) = 3-2x-x^2\), it meets all these conditions for continuity for every value in its domain, which is the set of all real numbers. This means the graph of \(f(x)\) is a single unbroken curve with no gaps or jumps — illustrating a perfect example of a continuous function. Because all these continuity conditions are satisfied for all real numbers, the function does not have any discontinuities.