Problem 5 - Algebra 2

Question

Find \([g \circ h](x)\) and \([h \circ g](x)\) $$ \begin{array}{l}{g(x)=2 x} \\ {h(x)=3 x-4}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understand Notation

The notation \( [g \circ h](x) \) represents the composite function \( g(h(x)) \). Similarly, \( [h \circ g](x) \) represents \( h(g(x)) \). Our task is to find both of these compositions.

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

Substitute \( h(x) = 3x - 4 \) into \( g(x) = 2x \) to get \( g(h(x)) \). We have:\[g(h(x)) = g(3x - 4) = 2(3x - 4) = 6x - 8\].

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

Substitute \( g(x) = 2x \) into \( h(x) = 3x - 4 \) to get \( h(g(x)) \). We have:\[h(g(x)) = h(2x) = 3(2x) - 4 = 6x - 4\].

Key Concepts

Function Notation

When learning about functions, it's crucial to become familiar with function notation. Function notation is a way to represent functions in a compact and understandable form. Typically, functions are written as letters like \( f(x) \), \( g(x) \), or \( h(x) \), where the letter represents the function, and \( x \) denotes the input or variable:

• The part inside the parentheses (e.g., \( x \)) is the input that the function acts upon.
• The letter before the parentheses represents the name of the function.
• The output is the value the function produces when you input \( x \) into it.

Understanding this is vital for grasping more complex concepts like composite functions. For instance, when you see \( g(x) = 2x \), you know for every input \( x \), the output will be \( 2 \times x \). Similarly, \( h(x) = 3x - 4 \) shows you how the function transforms an input \( x \) into \( 3x - 4 \). Function notation simplifies expressing complex mathematical operations with simple symbols.

Function Composition

Function composition is an operation that takes two functions and combines them into a single function. It's a critical concept in higher mathematics, especially in algebra.
In function composition, if you have two functions, say \( g(x) \) and \( h(x) \), you can build new functions by plugging one function into another. For example, the notation \( (g \circ h)(x) \) is read as "\( g \) of \( h \) of \( x \)," meaning you first apply \( h \) to \( x \), then \( g \) to the result:

• To find \( [g \circ h](x) = g(h(x)) \), you substitute \( h(x) \) into \( g(x) \). In our case, \( g(h(x)) = 2(3x - 4) = 6x - 8 \).
• Similarly, \( [h \circ g](x) = h(g(x)) \), you substitute \( g(x) \) into \( h(x) \). We calculate this as \( h(g(x)) = 3(2x) - 4 = 6x - 4 \).

Understanding how to perform function composition is fundamental because it helps you see how functions interact and combine to form new operations. It's a technique often used in calculus, computer science, and more advanced algebra courses.

Algebra 2 Functions

In Algebra 2, functions play a pivotal role in developing mathematical understanding. Functions are versatile tools that can model and solve a variety of problems.
Algebra 2 functions delve into more complex function operations and transformations:

• Composite functions like \( [g \circ h](x) \) are introduced, requiring students to combine and manipulate functions to understand relationships deeply.
• Polynomials, exponentials, and logarithmic functions are some advanced function types you explore.
• Learning involves not only computing outputs but also analyzing characteristics, such as domain and range.

Understanding functions in Algebra 2 means comprehensively exploring how they can be applied to model real-world scenarios, solve equations, and predict outcomes. As you progress, you'll see how powerful functions become in finding solutions to creative, practical problems.