Problem 41 - College Algebra

Question

Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=\sqrt[3]{x^{2}-9}$$

Step-by-Step Solution

Answer

The function can be expressed as a composition of functions \(f\) and \(g\) where \(f(x)=\sqrt[3]{x}\) and \(g(x)=x^{2}-9\), so \(h(x)=(f \circ g)(x)=\sqrt[3]{x^{2}-9}\).

Step 1: Select inner function g(x)

First, identify the inner function. We see that within the cube root, \(x^{2}-9\) is the inner part. We denote this part as function \(g(x)\). So, we have \(g(x)=x^{2}-9\).

Step 2: Select outer function f(x)

Next, we need to define the outer function that will act on the result of \(g(x)\). The outer function is the cube root operation. We denote this function as \(f(x)\). So, we have \(f(x)=\sqrt[3]{x}\).

Step 3: Express h(x) as a composition of f and g

Now that we have defined \(f(x)\) and \(g(x)\), \(h(x)\) can be expressed as composition of these two functions following the formula \((f \circ g)(x)\). The composition of \(f\) and \(g\) results in function \(h(x)\) = \(f(g(x))\).

Key Concepts

Inner and Outer Functions

Understanding inner and outer functions is like peeling an onion; there are layers to consider within complex functions. In the realm of algebra, inner and outer functions are two interdependent parts that make up composite functions.

The inner function is the one applied first; it's the 'inside layer' of our metaphorical onion. Any input given to a composite function will pass through the inner function before anything else. In our exercise example, the inner function is identified as a quadratic expression, represented by \(g(x) = x^2 - 9\).

Consequently, the outer function comes into play; it’s applied to the result given by the inner function. It’s the 'outside layer' that gives the final output after processing what the inner function produced. In this case, the exercise shows the outer function is the cube root operation, represented as \(f(x) = \sqrt[3]{x}\).

When a student struggles to identify these functions or sees the step-by-step solution as insufficient, it is helpful to visualize the function’s composition as layers, or to think of inserting the inner function output into the outer function just like fitting a key into a lock. The input of the outer function is always the output of the inner function.

Composition of Functions Algebra

The composition of functions in algebra involves combining two or more functions in a way that the output of one function becomes the input of another. This interplay creates a new function, often denoted as \((f \:circ g)(x)\) in algebraic terms.

In practice, to compose two functions, f and g, you apply them in succession, writing it as \(f(g(x))\), where g acts first and f second. The efficiency of this process is pivotal for conveying the correct application of function composition and can evoke a sense of simplicity, rather than complexity, to the students.

To assist in comprehension, it is beneficial to remind students that the order of composition matters significantly. The 'g inside f' notation is to be taken seriously, as reversing it to 'f inside g' produces a completely different function. Ensuring that students understand this order can be done by having them visualize or write out each step clearly, as seen in the sample exercise: \(h(x) = \sqrt[3]{x^{2} - 9} = f(g(x))\). A concrete grasp of the composition of functions is essential for their success in algebra and beyond.

Cube Root Function

Diving into the cube root function, one of the key actors in our example, it’s crucial to explain this operation in context. The cube root function, written as \(f(x) = \sqrt[3]{x}\), is the inverse operation of cubing a number. Just as squaring a number is multiplying it by itself twice, cubing is multiplying three times. Consequently, taking the cube root is finding the value that, when cubed, gives back the original number.

For students, the thought of cube roots might seem daunting at first, but providing the perspective that it’s just another form of 'undoing' an operation, similar to division being the 'undoing' of multiplication, can demystify it. The cube root function is peculiar because unlike the square root, which is only defined for non-negative numbers, cube root functions are defined for all real numbers, allowing for the inclusion of negative roots as well.

Using simple examples, like the cube root of 8 being 2, can illustrate this concept in familiar terms. Moreover, when visualizing the graph of a cube root function, point out its distinct s-shape, which can help students better recognize and understand its properties in geometric context.