Problem 39 - College Algebra

Question

Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=(3 x-1)^{4}$$

Step-by-Step Solution

Answer

The function h(x) = (3x-1)^4 can be expressed as a composition of two functions f(x) = x^4 and g(x) = 3x-1 such that h(x) = (f∘g)(x).

Step 1: Express h in terms of f and g

The function h(x) = (3x-1)^4 can be expressed as a composition of two functions, f and g. Here, we can choose function g which changes x to 3x-1, hence, g(x) = 3x - 1. Now, to get to h(x), the result of g(x) is raised to the power of 4. Therefore, the function f can be defined as f(x) = x^4.

Step 2: Verification of Composition

To verify that h(x) = (f∘ g)(x), we substitute g(x) into f. So \(f(g(x)) = f(3x - 1)\). Applying the definition of function f, we get \(f(g(x)) = (3x - 1)^4\), which matches the given function h(x). Hence, f(x) and g(x) are appropriate expressions that represent the function h(x) through the equation \(h(x) = (f \circ g)(x)\).

Step 3: Final Output

The decomposition of the function \(h(x) = (3x - 1)^4\) as a composition of two functions f and g is given as follows: \(f(x) = x^4\) and \(g(x) = 3x - 1\). Hence, the function h(x) can be expressed as \(h(x) = (f \circ g)(x)\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve operations like addition, subtraction, multiplication, division, and raising to a power. They are defined using algebraic operations on variables. In our exercise, the function \( h(x) \) is an algebraic function because it involves multiplying \( x \) by 3, subtracting 1, and then raising the result to the 4th power.

These types of functions form the backbone of algebra and are important because they describe many real-world situations. You can think of each operation as a step-by-step process that changes the input value (\( x \)) to produce an output.

• Example: For \( h(x) = (3x - 1)^4 \), components like multiplication and exponentiation are derived from consistent algebraic rules.
• Components: Algebraic operations include basic arithmetic transformations applied to variables.

Working with algebraic functions involves understanding how each operation influences the result. This understanding helps in transforming and manipulating expressions to solve equations or model phenomena.

Composite Functions

Composite functions occur when you combine two functions in sequence. For instance, if you have two functions \( f \) and \( g \), you can create a new function \( h \) by composing them, denoted as \( h(x) = (f \circ g)(x) \). Here, you first apply \( g \) to \( x \) and then apply \( f \) to the result of \( g(x) \). This concept is critical because it reflects a process of layered operations, where the output of one function becomes the input of another.

In our exercise, the composite function \( h(x) = (3x - 1)^4 \) can be seen as the composition of two simpler functions: \( g(x) = 3x - 1 \) and \( f(x) = x^4 \). This breakdown is practical in simplifying complex expressions by analyzing simpler parts.

• Process: Apply \( g \) first, transforming \( x \) by the rule \( g(x) = 3x - 1 \).
• Next Step: Apply \( f \) to \( g(x) \), resulting in \( f(g(x)) = (g(x))^4 = (3x - 1)^4 \).

Composite functions allow us to tackle complicated problems by breaking them down into straightforward steps, providing clarity and structure to mathematical processes.

Function Verification

Function verification ensures that a composite function performs exactly as expected. It involves substituting to confirm that the composition and the given target function are identical. This process verifies how well our choice of component functions recreates the original function.

By substituting \( g(x) = 3x - 1 \) into \( f(x) = x^4 \), we have \( f(g(x)) = (3x - 1)^4 \). This matches our given function \( h(x) = (3x - 1)^4 \). Thus, we verify that \( h(x) = (f \circ g)(x) \) is correct, ensuring validity.

• Steps to Verify: Substitute \( g(x) \) into \( f(x) \) and simplify.
• Confirmation: Observe if the simplified result matches the provided function.

Function verification is important as it assures correctness, helping eliminate errors in mathematical solutions, thereby solidifying confidence in functional composition and its application.