Problem 72 - Algebra 2

Question

Find \([g \circ h](x)\) and \([h \circ g](x)\) \(g(x)=2 x\) \(h(x)=3 x-4\)

Step-by-Step Solution

Answer

\([g \circ h](x) = 6x - 8\); \([h \circ g](x) = 6x - 4\)

Step 1: Understand Composition of Functions

To find the composition of two functions, say \( g \) and \( h \), we calculate \( [g \circ h](x) = g(h(x)) \) and \( [h \circ g](x) = h(g(x)) \). This means replacing \( x \) in one function with the expression from the other function.

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

To calculate \([g \circ h](x)\), substitute \( h(x) = 3x - 4 \) into \( g(x) = 2x \). This gives us: \( g(h(x)) = g(3x - 4) = 2(3x - 4) \). This simplifies to \( 6x - 8 \). Therefore, \([g \circ h](x) = 6x - 8\).

Step 3: Find [h \circ g](x)

To calculate \([h \circ g](x)\), substitute \( g(x) = 2x \) into \( h(x) = 3x - 4 \). This gives us: \( h(g(x)) = h(2x) = 3(2x) - 4 \). This simplifies to \( 6x - 4 \). Therefore, \([h \circ g](x) = 6x - 4\).

Key Concepts

Algebraic Expressions

Algebraic expressions are crucial building blocks in mathematics. In this context, they are expressions that combine numbers, variables, and operators (like addition or multiplication). For example, in the functions given, \( g(x) = 2x \) and \( h(x) = 3x - 4 \), both contain algebraic expressions. Here:

• \( 2x \) is an expression where \( x \) is being multiplied by 2.
• \( 3x - 4 \) is another expression, where 3 times \( x \) is subtracted by 4.

Understanding algebraic expressions is vital because they form the basis of creating functions and solving equations. This skill is essential in simplifying expressions, solving for variables, and in our example, performing function operations. Remember, the proper grasp of algebraic expressions also lays the groundwork for understanding more complex mathematical concepts as you progress.

Function Operations

Function operations involve different methods of combining and manipulating functions. In the exercise, we look at the operation known as 'composition of functions.' This is where one function is applied to the result of another function. Think of it like this:

• We have two functions, \( g(x) = 2x \) and \( h(x) = 3x - 4 \).
• To find \([g \circ h](x)\), you substitute \( h(x) \) into \( g(x) \), effectively applying \( g \) to the output of \( h \).
• Similarly, to find \([h \circ g](x)\), you substitute \( g(x) \) into \( h(x) \).

Function operations allow us to discover how different functions interact and work together. Through operations like addition, subtraction, multiplication, division, and composition, we gain deeper insights into the behavior of these mathematical entities. Being comfortable with these operations is key to mastering more advanced topics such as calculus.

Function Notation

Function notation is a way to symbolize and evaluate functions efficiently. In the problem, it's expressed as \( g(x) \) and \( h(x) \). This notation:

• Specifies the function's name - for example, \( g \) or \( h \).
• Shows the independent variable - usually \( x \).

Using function notation, we easily convey what happens to \( x \) in a function without writing out every step. It's also helpful for substituting expressions, as seen with \([g \circ h](x)\) or \( [h \circ g](x) \), indicating compositions.Function notation simplifies the process of evaluating and performing operations on functions. This makes it a crucial tool in mathematics, allowing mathematicians and students alike to effectively communicate complex ideas in a straightforward format. It's important to practice and become familiar with this notation to work smoothly with functions in algebra and beyond.