Problem 39 - Contemporary Precalculus

Question

Write the given function as the composite of two functions, neither of which is the identity function, as in Examples 6 and 7 . (There may be more than one way to do this.) $$f(x)=\sqrt[3]{x^{2}+2}$$

Step-by-Step Solution

Answer

Question: Identify two functions such that when composed, they yield the given function $$f(x)=\sqrt[3]{x^{2}+2}$$, making sure that neither of the functions is an identity function. Answer: The given function $$f(x)=\sqrt[3]{x^{2}+2}$$ can be written as the composite of two functions, $$u(x) = x^{2}+2$$ and $$v(u) = \sqrt[3]{u}$$, neither of which is the identity function.

Step 1: Find the inner function

The first step is to find a function that will perform the action inside the parentheses of the given function. That is, taking the square of the input and then adding 2. We define the inner function as: $$u(x) = x^{2}+2$$

Step 2: Find the outer function

The second step is to find a function that will perform the action of taking the cube root. We define the outer function as: $$v(u) = \sqrt[3]{u}$$

Step 3: Verify the composite and result

Finally, we'll verify that the composite of the two functions (i.e., \(v(u(x))\)) is equal to the given function: $$ v(u(x)) = \sqrt[3]{u(x)} = \sqrt[3]{x^{2}+2} = f(x) $$ The given function $$f(x)=\sqrt[3]{x^{2}+2}$$ can be written as the composite of two functions, neither of which is the identity function: $$f(x)=v(u(x))$$, where $$u(x) = x^{2}+2$$ and $$v(u) = \sqrt[3]{u}$$.

Key Concepts

Function Composition

Function composition is a mathematical operation where two functions are combined in a sequence to create a new function. Think of it like a two-step process in a factory: the output of the first function is fed into the second function, resulting in a final product.

In the context of our exercise, we look at this process in two stages: the inner function, and then the outer function. The inner function does an initial transformation of the input, and the outer function takes that result and does further processing.

If we want to consider an improvement for understanding, visualizing this process can be quite helpful. Imagine you're passing a value through two different machines. The first machine squares the number and adds two, and the second machine extracts the cube root of whatever comes out of the first one. The composition, denoted as \( v(u(x)) \), represents the sequence of operations from these two 'machines'. It's important to recognize that the order in which we compose functions matters, much like the order of operations in arithmetic.

Cube Root Function

The cube root function is the inverse operation of cubing a number; that is, if you have a number \( y \), finding a number \( x \), such that \( x^3 = y \). Symbolically, we represent the cube root of \( y \) as \( \sqrt[3]{y} \).

The most crucial aspect of a cube root function is understanding that it undoes the action of taking a number to the third power. If you imagine a number being expanded into a larger form by cubing, the cube root function compresses it back into its original form. This function can handle any real number, positive or negative, because cubing any real number results in a real number.

When teaching or improving the explanation of the cube root, use real-life scenarios, like the volume of a cube. For instance, if you know the volume and you want to find the length of the sides, the cube root will provide that. It helps connect abstract concepts to tangible examples.

Quadratic Function

A quadratic function is one where the highest degree of any term is two. The most general form of a quadratic function is \( f(x) = ax^2 + bx + c \), where \( a \) is not zero. Its graph is a parabola, and it can open upwards or downwards depending on the sign of \( a \). Quads stand for four, but in quadratics, we're talking about squaring—because the area of a square is a basic example of a quadratic expression.

Understanding the quadratic function is vital when dealing with real-world situations like projectile motion, or finding the area of a space when given only a side length. To improve understanding, when discussing quadratics, always bring up their visual representation - the parabola - and how it reflects the possible values of \( x \) that satisfy the function for any given \( y \). Also, touching on the real-world applications or integrating technology (like graphing calculators) can greatly enhance comprehension for students learning about quadratic functions.