Problem 27 - A Graphical Approach to College Algebra

Question

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt[3]{x+4}, g(x)=x^{3}+5$$

Step-by-Step Solution

Answer

Sum, difference, and product are valid for all real numbers. Division \(\frac{f}{8}\), compositions \(f \circ g\) and \(g \circ f\) are also all valid for all real numbers.

Step 1: Understanding the Functions

We are given two functions: \( f(x) = \sqrt[3]{x+4} \) and \( g(x) = x^3 + 5 \). Our goal is to perform operations and find their domains.

Step 2: Finding \((f+g)(x)\)

To find \((f+g)(x)\), add the functions: \( f(x) + g(x) = \sqrt[3]{x+4} + (x^3 + 5) = \sqrt[3]{x+4} + x^3 + 5 \).

Step 3: Finding \((f-g)(x)\)

Subtract \(g(x)\) from \(f(x)\): \( f(x) - g(x) = \sqrt[3]{x+4} - (x^3 + 5) = \sqrt[3]{x+4} - x^3 - 5 \).

Step 4: Finding \((fg)(x)\)

Multiply \(f(x)\) and \(g(x)\): \( f(x) \, g(x) = \sqrt[3]{x+4} \cdot (x^3 + 5) \).

Step 5: Domains for Sum, Difference, Product

All cubic roots and polynomial functions are defined for all real numbers, thus the domain for \( (f+g)(x) \), \( (f-g)(x) \), and \( (fg)(x) \) is \( \mathbb{R} \) (all real numbers).

Step 6: Finding \( \frac{f}{8} \)

Divide \(f(x)\) by 8: \( \frac{f}{8} = \frac{\sqrt[3]{x+4}}{8} \).

Step 7: Domain for \( \frac{f}{8} \)

Cubic root functions are defined for all real numbers, hence the domain of \( \frac{f}{8} \) is \( \mathbb{R} \) (all real numbers).

Step 8: Finding \( f \circ g \)

Substitute \(g(x)\) into \(f(x)\): \( f \circ g = f(g(x)) = \sqrt[3]{g(x) + 4} = \sqrt[3]{x^3 + 5 + 4} = \sqrt[3]{x^3 + 9} \).

Step 9: Domain for \( f \circ g \)

Since a cubic function and its transformations are defined for all real numbers, \( f \circ g \) has the domain \( \mathbb{R} \).

Step 10: Finding \( g \circ f \)

Substitute \(f(x)\) into \(g(x)\): \( g \circ f = g(f(x)) = (\sqrt[3]{x+4})^3 + 5 \). Note that \((\sqrt[3]{x+4})^3 = x + 4\), thus \( g \circ f = x + 4 + 5 = x + 9 \).

Step 11: Domain for \( g \circ f \)

The function simplifies to a linear function, \(x + 9\), which is defined for all real numbers, hence the domain is \( \mathbb{R} \).

Key Concepts

Domains of Functions

When dealing with functions, understanding their domain is crucial. The domain of a function is simply the set of all possible input values (usually "x" values) that will produce a valid output. For many basic functions, such as polynomial or cubic root functions, the domain is often all real numbers, denoted as \( \mathbb{R} \).

• For the sum, difference, and product of the functions \(f(x) = \sqrt[3]{x+4}\) and \(g(x) = x^3 + 5\), the domains are all real numbers because both the cubic root function and polynomial function are defined for every real number.
• The division operation also results in a domain of all real numbers, as long as it does not involve division by zero. In the exercise problem, dividing by a constant like 8 doesn’t introduce issues.
• Functions like composition can slightly change the domain, but in this case, both compositions \(f \circ g\) and \(g \circ f\) work out to have domains of all real numbers due to the types of operations involved and how these functions transform the inputs.

Thus, understanding the domain ensures that calculations remain valid and avoid any undefined behaviors.

Cubic Root Functions

Cubic root functions have interesting properties that differ slightly from other root functions. These functions take the form \(f(x) = \sqrt[3]{x + a}\). A defining characteristic of the cubic root function is that it accepts all real numbers as inputs. This is because you can find a real number whose cube is any given real number. Unlike square roots, cubic roots don’t require non-negative numbers inside the root.

• The function given in the problem \( \sqrt[3]{x + 4} \), for example, can take any value of \( x \), and it will yield a real number.
• This wide availability of \( x \) values means the function’s domain is simply \( \mathbb{R} \) (all real numbers).
• The transformation inside the root, \(+4\) in this case, just shifts the graph horizontally but doesn't restrict the domain.

Understanding these properties can greatly simplify problems involving cubic roots.

Polynomial Functions

Polynomial functions are a class of functions made up of variables raised to whole number exponents, such as \(f(x) = x^3 + 5\). These functions usually have very simple domains – often all real numbers - because there are no restrictions like square roots or divisions by variables (which could potentially restrict inputs).

• For example, in \( g(x) = x^3 + 5 \), any value of \( x \) can be plugged in, and it will return a valid output.
• The operations involved, including addition and multiplication by constants, don't limit their domain.
• Polynomials can describe a wide range of curves in the graph depending on the degree and coefficients, but the domain remains wide open.

Thus, polynomial functions provide flexibility while performing function operations or compositions.

Composition of Functions

The composition of functions involves plugging one function into another. This operation is denoted as \(f \circ g\), which means substituting the whole \(g(x)\) expression into \(f(x)\).

• This can change the nature of the resulting function and might affect its domain, but with cubic and polynomial functions, such changes don’t impose restrictions on real numbers as inputs.
• In this exercise, \(f \circ g\) resulted in a new function \(f(g(x)) = \sqrt[3]{x^3 + 9} \), and \(g \circ f\) led to \((\sqrt[3]{x+4})^3 + 5 = x + 9\), both of which are defined for all real numbers.
• The operation's requirements ensure the domain aligns with both input functions’ widest allowed sets of values, usually maintaining all real numbers.

This aspect of function composition enables more complex function behaviors and transformations, which are essential topics in calculus and advanced mathematics.