Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=x^{2 / 3}, \quad g(x)=x^{6}$$

Step-by-Step Solution

Answer

The composition functions \(f \circ g\) and \(g \circ f\) are both equal to \(x^4\). The domain of \(f \circ g\) is all real numbers (\(-\infty, +\infty\)). The graphs of \(f \circ g\) and \(g \circ f\) are identical, showing that \(f \circ g = g \circ f\).

Step 1: Compute the compositions

The first required task is to find \(f \circ g\) and \(g \circ f\). To find \(f \circ g\), replace \(x\) in \(f(x)\) with \(g(x)\): \(f(g(x)) = (x^6)^{2/3} = x^4\).For \(g \circ f\), replace \(x\) in \(g(x)\) with \(f(x)\): \(g(f(x)) = (x^{2/3})^6 = x^4\). So, \(f \circ g = g \circ f = x^4\).

Step 2: Determine the domain of f \circ g

The domain of a composite function is the set of all real numbers for which the composite function is defined. Since the function \(f(g(x)) = x^4\) is a polynomial, and polynomials are defined for all real numbers, the domain of \(f \circ g = x^4\) is all real numbers, or \((-\infty, +\infty)\) in interval notation.

Step 3: Graph f \circ g and g \circ f

To graph \(f \circ g\) and \(g \circ f\), use a graphing utility. Since both \(f \circ g\) and \(g \circ f\) resulted in \(x^4\), the graph of these composite functions will be the curve \(y = x^4\), which is a symmetric U-shaped curve with the lowest point at the origin (0,0).

Step 4: Determine whether f \circ g equals g \circ f

Once both composite functions have been graphed, compare the two graphs. Since the graphs for \(f \circ g\) and \(g \circ f\) overlap perfectly (they are both the curve \(y = x^4\)), it is correct to conclude that \(f \circ g = g \circ f\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To determine the domain, consider the expression inside the function and any constraints it might have.
Here are a few helpful tips:

• If a function involves a denominator, ensure it is not zero.
• For square roots or other even roots, the expression inside must be non-negative.
• Exponential and polynomial functions are generally defined for all real numbers, resulting in their domain being \((-\infty, \infty)\).

In our exercise, when we discussed \(f \circ g = x^4\), we realized it is a polynomial function. Since polynomial functions like this one are defined for all real numbers, its domain is the entire set of real numbers, \((-\infty, \infty)\), allowing any real number to be plugged in without restrictions.

Composite Function

A composite function is created when one function is applied to the result of another function. Mathematically, this is depicted as \((f \circ g)(x) = f(g(x))\). This means you first apply the function \(g\) to \(x\), and the result becomes the input for the function \(f\).
For example, in the exercise, we worked with \(f(x) = x^{2/3}\) and \(g(x) = x^6\):

• To find \(f \circ g\), substitute the expression for \(g(x)\) into \(f(x)\): \(f(g(x)) = (x^6)^{2/3} = x^4\).
• Similarly, for \(g \circ f\), substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = (x^{2/3})^6 = x^4\).

These compositions yield the same function, demonstrating that under certain conditions, \(f \circ g\) can equal \(g \circ f\), although generally, composition order matters.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A basic example of a polynomial is \(x^4\), which showcases the structure of adding and multiplying powers of \(x\).
Polynomial functions are characterized by:

• Simple expressions like constants (e.g., 7 or \(-5\)), variables to a power, or a combination.
• Being defined for all real numbers, unless specified otherwise.
• Smooth, continuous curves without gaps or sharp corners when graphed.

In the role of \(x^4\) seen in \(f \circ g\) and \(g \circ f\), it highlights how polynomial functions can result from composite functions. The simplicity and regularity of polynomials make them a fundamental building block in various mathematical analyses.