Problem 18 - College Algebra

Question

For the following exercises, use each set of functions to find \(f(g(h(x)))\) ). Simplify your answers. $$f(x)=x^{4}+6, g(x)=x-6, \text { and } h(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

The composed function is \((\sqrt{x} - 6)^4 + 6\).

Step 1: Understanding Composition of Functions

The problem requires us to find the composition of three functions, specifically \(f(g(h(x)))\). This means we'll start by finding the inner composition \(g(h(x))\), and then use that result in \(f(g(h(x)))\).

Step 2: Compose h(x) into g(x)

First, we need to substitute \(h(x)\), which is \(\sqrt{x}\), into \(g(x)\). So, substitute \(h(x)\) for \(x\) in the function \(g(x) = x - 6\):\[g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 6\]

Step 3: Compose g(h(x)) into f(x)

Next, use the result from Step 2, \(g(h(x)) = \sqrt{x} - 6\), and substitute it into \(f(x)\). The function \(f(x)\) is given by \(x^4 + 6\). Substitute \(g(h(x))\) for \(x\):\[f(g(h(x))) = f(\sqrt{x} - 6) = (\sqrt{x} - 6)^4 + 6\]

Step 4: Simplify the Expression

While the expression \((\sqrt{x} - 6)^4 + 6\) is the composition, further simplification can involve expanding \((\sqrt{x} - 6)^4\) using the binomial theorem, but it is an extensive algebraic process not required unless more specific simplification is requested. Thus, the composed and simplified expression is:\[f(g(h(x))) = (\sqrt{x} - 6)^4 + 6\]

Key Concepts

Algebraic Functions

Algebraic functions are powerful tools that allow us to explore relationships between different quantities. They are functions expressed using polynomial, rational, and even sometimes radical expressions. In our exercise, we have three algebraic functions to consider:

• \(f(x) = x^4 + 6\): This is a polynomial function where the variable \(x\) is raised to the fourth power and added to 6. It represents a parabola with a high degree and smooth curve.
• \(g(x) = x - 6\): A simple linear function that decreases the input \(x\) by 6. Graphically, this represents a straight line with a slope of 1.
• \(h(x) = \sqrt{x}\): A radical function involving the square root of \(x\). It represents only the positive branch of the parabola \(y^2 = x\).

Understanding these functions, we can appreciate how they transform any input \(x\) into outputs through their specific rules. This sets the stage for deeper exploration in function composition, where we create new functions by combining existing ones.

Function Operations

Function operations involve performing algebraic computations on functions to find new outcomes. This consists of processes such as addition, subtraction, multiplication, division, and, as in the given problem, composition. Function composition is a critical operation where we apply one function to the results of another, effectively chaining functions together.

Here's how the composition works in our problem, using three functions:

• Start by applying \(h(x) = \sqrt{x}\) to any input \(x\). This serves as the first transformation.
• Then, \(g(x)\), which is \(x - 6\), takes this result; the composition \(g(h(x))\) subtracts 6 from \(\sqrt{x}\).
• Finally, take the result from \(g(h(x))\) and apply the polynomial function \(f(x) = x^4 + 6\) to compute \(f(g(h(x)))\).

By understanding these individual operations, students can predict complex transformations by following each step methodically. This step-by-step approach is essential for solving more intricate algebraic problems and developing analytical skills.

Mathematical Expressions

Mathematical expressions like \((\sqrt{x} - 6)^4 + 6\) can initially seem daunting, but they have a defined structure that makes them solvable with patience and precision. Here are some tips for simplifying such expressions:

• Identify the expression's individual components: In the expression \((\sqrt{x} - 6)^4 + 6\), identify \(x\) within the square root and understand that 6 is constant.
• Use the binomial theorem: This theorem helps expand expressions of the form \((a + b)^n\). Though tedious for manual calculations, it illustrates the fundamental nature of polynomial expansion.
• Recognize constant terms: Simplifications often both maintain constants and simplify repeated patterns through factorization and transformation.

With practice, expressing complex equations reveals underlying simplicity, providing insights into the principles of algebra. It fosters an appreciation for mathematical beauty and builds confidence in managing a wide range of expressions.