Problem 76 - College Algebra Essentials

Question

Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=(2 x-5)^{3}$$

Step-by-Step Solution

Answer

The given function \(h(x)=(2 x-5)^{3}\) can be expressed as a composition of two functions. The function \(g(x)\) is \(2x - 5\), and the function \(f(x)\) is \(x^{3}\). Therefore, \(h(x)=(f \circ g)(x)\).

Step 1: Identify Function \(g(x)\)

Firstly, examine \(h(x)=(2 x-5)^{3}\) and determine which operation could be considered an operation on \(x\), this operation will be \(g(x)\). In this case, \(g(x)\) appears to be the operation inside the brackets which is \(g(x) = 2x - 5\)

Step 2: Identify Function \(f(x)\)

Secondly, consider what is done with the result of \(g(x)\) to get from \(g(x)\) to \(h(x)\). After \(g(x) = 2x - 5\), the next operation is a cube, so \(f(x)\) is the cubic function, meaning \(f(x) = x^{3}\).

Step 3: Test Function Composition \(h(x)=(f \circ g)(x)\)

To confirm that \(f(g(x))\) equals \(h(x)\), substitute \(g(x)\) into \(f(x)\). This results in \(f(g(x)) = (2x - 5)^3\) which is equivalent to \(h(x)\). Thus \(h(x) = (f \circ g)(x)\)

Key Concepts

Functions in Algebra

In algebra, a function is a relationship between two sets that assigns each input exactly one output. Functions are often written as commands for transformations we can perform with numbers. For example, when we talk about a function \( h(x) = (2x - 5)^3 \), we are describing a rule that transforms any number \( x \) into the result of calculating \((2x - 5)^3\). In this context, composing functions means creating a new function by applying one function to the results of another function.
In our exercise, we demonstrate function composition by using two functions, \( f(x) \) and \( g(x) \), to reinterpret \( h(x) \). This approach allows us to build more complex functions from simpler ones, making it a powerful tool for solving various algebraic problems. It simplifies problems by breaking down a complex transformation into two more manageable parts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators that together represent a particular value or set of values. In the expression \( h(x) = (2x - 5)^3 \), we can observe different components:

• **Numbers**: These are constants in the expression, like 2 and 5.
• **Variables**: The letter \( x \) represents an unknown value.
• **Operators**: The minus sign (-) and exponentiation (\(^3\)).
• **Grouping Symbols**: Parentheses are used to group terms into sub-expressions.

Understanding these elements helps us to manipulate and transform expressions, especially in function composition. Here, by recognizing \( g(x) = 2x - 5 \) and \( f(x) = x^3 \), the problem demonstrates translating complex expressions into simple components.

Mathematical Operations

Mathematical operations form the foundation of algebra and involve four basic types: addition, subtraction, multiplication, and division. These operations help us solve problems and understand expressions. In the context of composing functions, as seen with \( h(x) = (2x - 5)^3 \), the primary operations involved are multiplication and exponentiation.
By examining the function \( g(x) = 2x - 5 \), we focus on subtraction and multiplication. The expression takes \( x \), multiplies it by 2, then subtracts 5, representing transformation operations.
For \( f(x) = x^3 \), exponentiation is introduced, raising the output of \( g(x) \) to the power of three, adding layers to our interpretation. Understanding mathematical operations in functions allows students to efficiently break down and solve more complex algebraic challenges.