Question

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\sqrt{1-x^{2}} $$

Short answer

The inside function is \(u=g(x) = x^{2}\) and the outside function is \(y=f(u) = \sqrt{1-u}\).

Step-by-step solution

Identify the Innermost Expression

The first thing to do is find the innermost operation. Looking at the function \(y=\sqrt{1-x^{2}}\), the innermost operation is the part within the square root function, which is \(1-x^{2}\). Thus, \(u=x^{2}\). An expression of \(u\) in terms of \(x\) is important because it isolates the inner function.

Recognise the Outside Function

The next step is to identify the outer function, which is operated on the result of the inner function. The outside function is the operation that's being performed on \(u\) – in this case, taking the square root of \(1-u\). Thus, \(y=f(u)=\sqrt{1-u}\).

Check the Identification

To ensure that you've correctly identified the inner and outer functions, substitute \(u\) back into the outer function. If \(u=x^{2}\) and \(y=\sqrt{1-u}\), substituting gives \(y=\sqrt{1-x^{2}}\), which is your original function, confirming that the steps are correct.

Key concepts

Inner and Outer Functions

Understanding the concepts of inner and outer functions is crucial when dealing with composite functions. A composite function is akin to nesting one function within another. To identify these components, one must dissect the function, much like peeling an onion layer by layer.

The inner function is typically the part that is 'inside' another function, which is applied first. In the expression y = \( \sqrt{1-x^{2}} \), the inner function is represented by u = g(x), where g(x) is 1-x^{2}. This inner function is the core or the initial operation on x that must be performed before applying the subsequent outer function.

The outer function, y = f(u), is the function that takes the result of the inner function as its input. In our example, once we've calculated u, the outer function f(u) is applied, which in this case is the square root function. Recognizing these distinctions is essential for deconstructing and reconstructing composite functions, which is a foundational skill in higher-level math courses.

Function Composition

Function composition is the process of combining two or more functions in such a way that the output of one function becomes the input of the next. It's a bit like passing a baton in a relay race: the output of the one function is smoothly handed off to the input of another.

In mathematical notation, this concept is captured by the formula y = f(g(x)). Here, f and g represent two distinct functions, with g serving as the first function applied to x, and f taking the result of g(x) as its input. The order matters tremendously—reversing f and g would result in a potentially very different function.

To effectively work with function composition, it's beneficial to take a step-by-step approach, first solving for the inner function and then applying the outer function. A common mistake is to overlook this sequential nature and attempt to solve both functions simultaneously. By understanding function composition, students can build complex functions from simpler ones, an ability that's extremely useful in both calculus and algebra.

Algebraic Functions

Moving on from the composition of functions, let's delve into algebraic functions. These functions are made up of operations commonly found in algebra, including addition, subtraction, multiplication, division, and nth roots. When we express functions using these operations and variables, we open the door to a vast array of function types, each with its own characteristics and behaviors.

Consider our composite function y = \( \sqrt{1-x^{2}} \). This function is algebraic because it involves the subtraction of x^{2} from 1, followed by the computation of a square root. Algebraic functions like this one are essential in understanding the behavior of quantities and their relations, serving as a tool to describe and analyze real-world phenomena.

Familiarity with algebraic functions equips students with the ability to solve equations, model situations, and predict outcomes within various scientific fields. It's the bedrock upon which more advanced mathematical concepts are built, making its mastery a pivotal aspect of any mathematics education curriculum.