Problem 75 - College Algebra Essentials

Question

In Exercises \(75-82,\) 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 \(h(x)\) can be expressed as a composition of two functions \(f\) and \(g\) where \(f(x)=x^4\) and \(g(x)=3x-1\).

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

Firstly we determine \(g(x)\). Here, \(g(x)\) should correspond to the inner functioning of \(h(x)\). \(g(x)=3x-1\). This takes an input, multiplies it by 3, and subtracts 1.

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

Then we determine \(f(x)\). This operates on the result of \(g(x)\), namely, it raises the result of \(g(x)\) to the power of 4. Therefore, \(f(x)=x^4\).

Step 3: Express \(h(x)\) as the Composition of \(f\) and \(g\)

Finally, we can express \(h(x)\) as a composition of \(f(x)\) and \(g(x)\). We write \(h(x)=f(g(x))\), which translates to \(h(x)=(3x-1)^4\).

Key Concepts

Understanding Algebraic Functions

Algebraic functions are expressions that involve sums, products, divisions, and powers of variables. In this type of function, the relationships between variables are constructed using arithmetic operations. For instance, the function given in the exercise, \( h(x) = (3x - 1)^4 \), is an algebraic function, as it involves subtraction and exponentiation.
One key feature of algebraic functions is their ability to represent complex relationships within concise mathematical expressions. They form the foundation for many mathematical processes, including function compositions, which allow for more intricate calculations and manipulations of mathematical relationships.
With algebraic functions, we can easily manipulate linear, quadratic, cubic, and higher-degree polynomials. This inherent versatility makes them essential in various fields from engineering to economics. In practice, understanding how to break down and manipulate these functions is crucial for solving many real-world problems.

Delving into Function Composition

The composition of functions is a fundamental concept in mathematics, especially within algebra. It involves the operation of combining two functions where the output of one function becomes the input of another. This process is visually expressed as \( (f \circ g)(x) = f(g(x)) \).
Consider the problem statement where \( h(x) = (3x - 1)^4 \). Here, we are tasked with expressing \( h(x) \) as \( f(g(x)) \).
- We start by identifying \( g(x) \) as the inner function, \( g(x) = 3x-1 \). This function modifies the input first.
- Next, \( f(x) \) is identified as the outer function, which receives the output of \( g(x) \) and processes it further by raising it to the fourth power, making \( f(x) = x^4 \).
Through function composition, we create a seamless chain of operations, enabling us to simplify complex tasks by breaking them into more manageable steps. This process is essential for ensuring calculations are both precise and simplified.

Engaging with Student Exercises

Exercises like these are pivotal in helping students build a strong foundation in mathematics. By practicing function composition, students gain a deeper understanding of how complex functions can be broken down into simpler components. This not only aids in solving mathematical problems but also enhances cognitive skills such as problem-solving and logical reasoning.
When tackling these exercises, students should:

• Identify the inner and outer functions clearly.
• Rewrite complex expressions into a series of simpler operations.
• Check their work by substituting values to verify the composition correctly matches the original function.

Such exercises are designed to incrementally build your skills, preparing you for more advanced mathematical concepts in the future. Confidence grows with practice, making these foundational exercises crucial for academic success.