Problem 27 - A Graphical Approach to Precalculus with Limits

Question

For each pair of fimetions, (a) find ( \(f+g)(x),(f-g)(x),\) and \((f g)(x) ;\) (b) give the domains of the functions in part (a); (c) find \(\frac{t}{x}\) and give its domain; (d) find \(f \circ g\) and give is 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

All composite and arithmetic functions are defined for all real numbers.

Step 1: Find \((f+g)(x)\)

To find \((f+g)(x)\), simply add the expressions for \(f(x)\) and \(g(x)\). Therefore, \((f+g)(x) = \sqrt[3]{x+4} + (x^3 + 5)\).

Step 2: Find \((f-g)(x)\)

To find \((f-g)(x)\), subtract the expression for \(g(x)\) from that of \(f(x)\). Thus, \((f-g)(x) = \sqrt[3]{x+4} - (x^3 + 5)\).

Step 3: Find \((f \cdot g)(x)\)

To find \((f \cdot g)(x)\), multiply the expressions for \(f(x)\) and \(g(x)\). Therefore, \((f \cdot g)(x) = \sqrt[3]{x+4} \cdot (x^3 + 5)\).

Step 4: Determine the Domains of (a)

The domain for \((f+g)(x)\), \((f-g)(x)\), and \((f \cdot g)(x)\) is all real numbers. Both \( \sqrt[3]{x+4} \) and \(x^3 + 5\) are defined for all real \(x\).

Step 5: Find \(\frac{f}{g}(x)\)

To find \(\frac{f}{g}(x)\), divide \(f(x)\) by \(g(x)\). Therefore, \(\frac{f}{g}(x) = \frac{\sqrt[3]{x+4}}{x^3 + 5}\).

Step 6: Determine the Domain of \(\frac{f}{g}(x)\)

For \(\frac{f}{g}(x)\), exclude any \(x\) values that make \(g(x) = 0\). Since \(x^3 + 5 = 0\) when \(x = -\sqrt[3]{5}\), the domain is all real numbers except \(-\sqrt[3]{5}\).

Step 7: Find \(f \circ g(x)\)

To find \(f \circ g(x)\), substitute \(g(x)\) into \(f(x)\). Hence, \(f \circ g(x) = \sqrt[3]{(x^3 + 5) + 4} = \sqrt[3]{x^3 + 9}\).

Step 8: Determine the Domain of \(f \circ g(x)\)

The expression \(\sqrt[3]{x^3 + 9}\) is defined for all real numbers, so the domain of \(f \circ g(x)\) is all real numbers.

Step 9: Find \(g \circ f(x)\)

To find \(g \circ f(x)\), substitute \(f(x)\) into \(g(x)\). Thus, \(g \circ f(x) = (\sqrt[3]{x+4})^3 + 5 = x + 4 + 5 = x + 9\).

Step 10: Determine the Domain of \(g \circ f(x)\)

For \(g \circ f(x) = x + 9\), which is a linear function, the domain is all real numbers.

Key Concepts

Function Composition

Function composition involves combining two functions to make a new function. It's like connecting two machines in series: the output of one becomes the input of the other. For instance, if you have functions \( f(x) \) and \( g(x) \), the composition \( f \circ g(x) \) means you plug \( g(x) \) into \( f(x) \). In our exercise, this was \( f \circ g(x) = \sqrt[3]{x^3 + 9} \).

This process is uniquely powerful because it allows us to transform inputs in multiple steps without combining all operations into a single massive function. Additionally, exploring the reverse composition \( g \circ f(x) \) focuses on using \( f(x) \) first, followed by \( g(x) \). In this case, \( g \circ f(x) = x + 9 \), showing how compositions can lead to a dramatically different result based on order.

Function Domains

The domain of a function is the set of allowed inputs, where the function produces a real output. Each function can have a distinct domain influenced by operations within its formula.

• For computations like \( (f+g)(x) \), \( (f-g)(x) \), and \( (f \cdot g)(x) \), both functions \( f \) and \( g \) must be defined at \( x \). Here, they are defined for all real numbers due to the nature of cube root and polynomial.
• In the division function \( \frac{f}{g}(x) \), the domain excludes values where the denominator is zero. So the domain here is all real numbers except \( x = -\sqrt[3]{5} \).
• For compositions like \( f \circ g(x) \) and \( g \circ f(x) \), the domains are determined by the innermost functions' domains and any additional restrictions from the outer function.

Algebraic Expressions

Algebraic expressions form the backbone of functions. They are combinations of numbers, variables, and operations like addition and multiplication. Understanding the manipulation of these expressions is crucial for finding the results of function operations.

For example, combining two expressions \( f(x) = \sqrt[3]{x+4} \) and \( g(x) = x^3 + 5 \) can result in various new forms when added, subtracted, or composed. The expressions instruct us on how to rearrange and simplify operations. By adding, subtracting, or multiplying, you create new functions such as \( (f+g)(x) \) = \( \sqrt[3]{x+4} + (x^3 + 5) \) or the more complex \( \frac{f}{g}(x) \).

When functions like these combine, it's critical to manage the algebra to understand each step. This comprehension enhances the ability to explore further complex functions operations.