Question

Discuss the continuity of the composite function \(h(x)=f(g(x))\). $$ \begin{array}{l} f(x)=x^{2} \\ g(x)=x-1 \end{array} $$

Short answer

The composite function \(h(x) = f(g(x)) = (x - 1)^2\) is continuous for all real numbers.

Step-by-step solution

Consider the individual functions

The first function is \(f(x) = x^2\), a simple quadratic function, which is continuous for all real numbers. The second function is \(g(x) = x - 1\), a linear function, which is also continuous for all real numbers.

Construct the composite function

The composite function \(h(x)\) is a combination of the two functions, such that \(h(x) = f(g(x))\). Given that \(f(x) = x^2\) and \(g(x) = x - 1\), substitute \(g(x)\) into \(f(x)\) to create the composite function: \(h(x) = f(g(x)) = ((x - 1)^2)\).

Analyze the continuity of the composite function

The composite function \(h(x) = (x - 1)^2\) is a quadratic function, which, similar to \(f(x) = x^2\), is continuous for all real numbers.

Key concepts

Composite Functions

When learning about composite functions, we consider the process of combining two functions into one by applying one function to the result of another. The composite function is denoted as h(x) = f(g(x)), where g(x) is a function applied to x, and then f(x) is applied to the result of g(x). This chaining together forms the foundation of composite functions.

Understanding the behavior of composite functions requires examining the continuity and differentiability of both functions involved. If both functions, f(x) and g(x), are continuous over a certain interval, then their composition will also be continuous over that interval. This property is pivotal when solving calculus problems, as it simplifies the analysis of more complex functions created from simpler ones.

In our exercise, we created a composite function by substituting g(x) = x - 1 into f(x) = x^2, leading to the new function h(x) = (x - 1)^2. This procedure illustrates the creation of a composite function, reinforcing the concept that the output of g(x) becomes the input for f(x).

Quadratic Function

A quadratic function is a second-degree polynomial, typically represented in the standard form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Quadratic functions have distinctive properties, such as the shape of their graphs being parabolas, and they have a range of applications in physics, engineering, and economics.

The graph of a quadratic function opens upwards if a > 0 and opens downwards if a < 0. The vertex, which is the highest or lowest point of the parabola, and the axis of symmetry, which is a vertical line that bisects the parabola, are important characteristics that help understand the function's behavior.

In our context, both the base function f(x) = x^2 and the composite function h(x) = (x - 1)^2 are quadratic functions. The continuity and shape of these functions are straightforward to analyze because they are defined for all real numbers, and their graphs have no breaks, jumps, or holes.

Continuous Functions

The concept of continuous functions is fundamental in calculus, describing functions that are unbroken or seamless over their domains. Essentially, a function is continuous at a point if we can draw its graph at that point without lifting the pencil from the paper. Formally, a function f(x) is continuous at a point a if the limit as x approaches a is equal to f(a).

A function is continuous over an interval if it is continuous at every point within that interval. This means that linear functions, like g(x) = x - 1, and quadratic functions, like f(x) = x^2, are continuous over all real numbers. They don't exhibit any abrupt changes in value and are predictable without any gaps.

In dealing with continuity of composite functions, we can confidently say, based on the underlying functions' continuity, that h(x) = (x - 1)^2 is continuous everywhere. This understanding is not only crucial for solving calculus problems but also for understanding how functions behave and how they can be manipulated in various mathematical contexts.