Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=x^{1 / 4}, \quad g(x)=x^{4}$$

Step-by-Step Solution

Answer

The domain for \(f(x)\), \(g(x)\), and \(f \circ g\) is all real numbers.

Step 1: Determine the domain of \(f(x)\)

The function \(f(x)\) is \(x^{1 / 4}\). The fourth root of a number is defined for all real numbers in this case because the fourth root of a number is the unique real number which, when multiplied by itself four times, equals the original number. Thus, for \(f(x)\), any real number falls under the domain. So the domain of \(f(x)\) is all real numbers.

Step 2: Determine the domain of \(g(x)\)

The function \(g(x)\) is \(x^{4}\). For the fourth power of a number, the domain is also all real numbers since you can multiply any real number by itself four times. Therefore, the domain of \(g(x)\) is all real numbers.

Step 3: Determine the domain of the composite function

The composite function \(f \circ g\) means \(f(g(x))\). To find its domain, first substitute \(g(x)\) into \(f\). We get \(f(g(x)) = f(x^{4}) = (x^{4})^{1 / 4}\), simplifying this we get \(f(g(x)) = x\). Here, x can be any real number. Hence, the domain of \(f \circ g\) is all real numbers.

Step 4: Verify results using a graphing utility.

Use a graphing utility to plot the functions \(f(x)\), \(g(x)\), and \(f \circ g\) separately. By analyzing the plots, one can see that in each case, the function line continues indefinitely in both directions on the x-axis, demonstrating that the domain of each function is all real numbers.

Key Concepts

Composite functions

A composite function is a function that results from applying one function to the results of another function. It is often represented as \((f \circ g)(x)\), which means you first apply function \(g\) to \(x\), then apply function \(f\) to the result of \(g(x)\). Think of it like a two-step process. First, you transform your input using function \(g(x)\), which is the inner function, and then use this output as the input for the outer function \(f\).

• Notation: \((f \circ g)(x)\)
• Process: Compute \(g(x)\) and then \(f(g(x))\)

This approach is particularly useful for simplifying expressions and solving complex problems where direct computation might be difficult.
When determining the domain of a composite function, it's important to consider both functions' domains. You must ensure thatthe output from the first function \(g(x)\) is valid input for the second function \(f\). For the example we've explored, the composite function simplifies to \(f(g(x)) = x\). Since the final expression is just \(x\), the domain is all real numbers.

Graphing utility

A graphing utility is a tool, such as software or a graphing calculator, that displays the graph of a mathematical function. It offers a visual representation of the function, which is extremely handy in verifying the domain and behavior of functions. Using graphing utilities can aid in:

• Verifying calculated domains visually
• Exploring curves and intersections
• Solving equations graphically

For the functions \(f(x) = x^{1/4}\), \(g(x) = x^{4}\), and their composite, you can use a graphing utility to plot them individually. By inspecting these graphs, you will notice that each continues indefinitely along the x-axis, affirming that their domains include all real numbers.
Graphing utilities are incredibly beneficial for those visual learners out there, as they transform abstract equations into images that can be analyzed easily.

Polynomial functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers and multiplied by coefficients. They can have constants, variables, and exponents that are non-negative integers. The simplest polynomial is a linear equation like \(f(x) = x\), but they can also be quadratic, cubic, or higher degree functions. Characteristics of polynomial functions include:

• They are continuous and smooth
• Can have multiple terms (monomial, binomial, etc.)
• Have an infinite domain of all real numbers
• Their graphs have predictable shapes such as parabolas or higher degree curves

The function \(g(x) = x^4\) is a polynomial function. Since polynomial functions are not restricted by operations that might limit domains, such as division or square roots of negative numbers, \(g(x)\) is defined for every real number, leading to a domain of all real numbers. Polynomials are convenient due to their uncomplicated domains and reliable graph patterns.