Problem 28 - Algebra Form and Function

Question

Find (a) \(\quad f(g(x))\) (b) \(g(f(x))\) $$ f(x)=x^{3}+1 \text { and } g(x)=\sqrt{x} $$

Step-by-Step Solution

Answer

Answer: (a) f(g(x)) = √(x^3) + 1 (b) g(f(x)) = √(x^3 + 1)

Step 1: Find f(g(x))

To find f(g(x)), replace x in the f(x) function with g(x): $$ f(g(x)) = (g(x))^3 + 1 $$ Now, substitute the expression for g(x) in the above equation: $$ f(g(x)) = (\sqrt{x})^3 + 1 $$

Step 2: Simplify f(g(x))

Simplify the expression for f(g(x)): $$ f(g(x)) = \sqrt{x^3} + 1 $$ So we have found f(g(x)): $$ f(g(x)) = \sqrt{x^3} + 1 $$

Step 3: Find g(f(x))

To find g(f(x)), replace x in the g(x) function with f(x): $$ g(f(x)) = \sqrt{f(x)} $$ Now, substitute the expression for f(x) in the above equation: $$ g(f(x)) = \sqrt{x^3 + 1} $$

Step 4: Simplify g(f(x))

g(f(x)) is already simplified, so we have found g(f(x)): $$ g(f(x)) = \sqrt{x^3 + 1} $$ Therefore: (a) f(g(x)) = $$ \sqrt{x^3} + 1$$ (b) g(f(x)) = $$ \sqrt{x^3 + 1}$$

Key Concepts

Nested Functions

Function composition is a crucial concept in mathematics, which involves applying one function to the results of another. This is often referred to as nesting functions. Think of it like a process where you first use one function and then another. For example, if you have two functions, \( f(x) \) and \( g(x) \), the nested function \( f(g(x)) \) means you first apply \( g \) to \( x \), and then \( f \) to the result. This layering of functions can be visualized as putting one function inside another, similar to a set of Russian dolls, where each layer adds or changes something in the operation.
Some tips to remember about nested functions include:

• Identify the inner function and solve it first
• Use the result of the inner function as the input for the outer function
• Substitute carefully to avoid mistakes

This method can be applied to a wide range of mathematical problems, from simple arithmetic operations to complex algebraic equations.

Algebraic Functions

Algebraic functions are built from constants, variables, and algebraic operations such as addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In our specific exercise, the function \( f(x) = x^3 + 1 \) is an example of an algebraic function. It involves taking a variable \( x \), raising it to the power of 3, and adding 1. Such functions are essential in algebra as they form the building blocks for more complex expressions and equations.
Some key features of algebraic functions include:

• They can be both simple and complex
• They play a vital role in expressing relationships between variables
• They are foundational for calculus and advanced mathematics

Understanding how to manipulate and work with algebraic functions is crucial for successfully solving equations and math problems.

Square Root Function

The square root function, often represented as \( g(x) = \sqrt{x} \), is a specific type of function involving the operation of finding the square root of a number. It's an example of a non-linear operation because the rate of change of the function is not constant. The square root operation is common in many mathematical contexts, especially in solving quadratic equations and optimizing problems.
Here are some important points about square root functions:

• The domain is all non-negative numbers because the square root of a negative number is not defined in the set of real numbers
• They often appear in the form \( \sqrt{x} \), implying a function with a gentle increasing curve
• They are commonly involved in geometry and physics problems

Understanding the square root function's properties and applications is essential, as it frequently comes up in various math scenarios. It allows us to solve equations where the variable is squared, and is also linked with the geometric concept of finding the side of a square with a given area.