Problem 30 - Algebra 2

Question

Find \([g \circ h](x)\) and \([h \circ g](x)\) $$ \begin{array}{l}{g(x)=x+2} \\ {h(x)=x^{2}}\end{array} $$

Step-by-Step Solution

Answer

[g \circ h](x) = x^2 + 2,\ [h \circ g](x) = x^2 + 4x + 4.

Step 1: Understanding the Problem

We need to find the composition of functions \(g \circ h\)(x)\ and \(h \circ g\)(x). This means we substitute one function into the other. For \(g \circ h\), we will substitute \(h(x)\) into \(g(x)\), and for \(h \circ g\), we will substitute \(g(x)\) into \(h(x)\).

Step 1: Find \\(g \\circ h\\)(x)

Substitute \(h(x)\) into \(g(x)\). First, note that \(h(x) = x^2\). Therefore, \(g(h(x)) = g(x^2)\). Substitute \(x^2\) into \(g(x) = x+2\) to get \(g(x^2) = x^2 + 2\).

Step 2: Find \\(h \\circ g\\)(x)

Substitute \(g(x)\) into \(h(x)\). Given \(g(x) = x + 2\), set \(h(g(x)) = h(x + 2)\). Substitute \(x + 2\) into \(h(x) = x^2\) to get \((x + 2)^2\). Simplify \((x + 2)^2 = x^2 + 4x + 4\).

Step 4: Summary of Results

We have found that \(g \circ h(x) = x^2 + 2\) and \(h \circ g(x) = x^2 + 4x + 4\).

Key Concepts

Mathematical Functions

Mathematical functions are like machines that take an input, do something specific with it, and give an output. They are usually expressed as \(f(x)\), where \(x\) is a variable, and can represent all kinds of operations. Functions are pervasive in mathematics because they help us understand how different quantities relate to one another.

Consider the functions given in the exercise: \(g(x) = x+2\) and \(h(x) = x^2\). Here, \(g(x)\) adds 2 to any input \(x\), while \(h(x)\) squares the input. These simply defined operations demonstrate the core idea of a function: modify input to produce a specific output. Functions serve as the building blocks in advanced mathematical concepts and real-world applications.

A basic understanding of functions lays the groundwork for more complicated operations, such as function composition, which involves creating a new function by combining existing ones.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators that represent specific values or relationships. They often form the basis for equations and functions. Basic elements include constants like numbers and variables, which stand for unknown or changeable values. Operators like addition, subtraction, multiplication, and division join these elements together.

In the exercise, each function \(g(x) = x+2\) and \(h(x) = x^2\) is an algebraic expression. An expression can be as simple as a single number or variable or more complex like the squared term \(x^2 + 4x + 4\) found in the composition \(h \circ g(x)\). It is important to understand how to manipulate these expressions: simplify, expand, and substitute, as these skills are essential for solving algebraic equations and performing function operations.

• Terms: Parts of expressions separated by addition or subtraction.
• Coefficients: Numbers multiplied by variables, like the 4 in \(4x\).
• Degree: The highest power of the variable in the expression.

Being comfortable with algebraic expressions allows us to work easily with functions and develop solutions to broader mathematical problems.

Function Operation

Function operation is a crucial aspect of working with functions, involving processes like addition, subtraction, multiplication, division, and especially function composition. Function composition, often represented by \((f \circ g)(x)\), means inserting one function into another, much like nesting machines. It's a way to layer formulas and create new outputs.

In the exercise, to find \((g \circ h)(x)\) and \((h \circ g)(x)\), we substitute one function result into another, effectively combining their operations. For \((g \circ h)(x)\), \(h(x) = x^2\) is input into \(g(x) = x + 2\), resulting in the output \(x^2 + 2\). Conversely, for \((h \circ g)(x)\), \(g(x) = x + 2\) is input into \(h(x) = x^2\), producing \(x^2 + 4x + 4\) after simplification.

Steps to perform function composition:

• Identify the inside and outside functions.
• Substitute the inside function into the outside function.
• Simplify the resulting expression if necessary.

Understanding function operations expands our ability to solve complex problems and apply mathematical concepts in real-life scenarios, capitalizing on the synergy of combined function effects.