Problem 18 - College Algebra

Question

Use each set of functions to find \(f(g(h(x)))\). Simplify your answers. \(f(x)=x^{4}+6, g(x)=x-6,\) and \(h(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

\(f(g(h(x))) = x^2 - 24x^{3/2} + 216x - 858\).

Step 1: Find h(x)

The function given for \(h(x)\) is \(h(x) = \sqrt{x}\). This is the innermost function that will be applied first in the composite function \(f(g(h(x)))\).

Step 2: Find g(h(x))

Next, substitute \(h(x)\) into \(g(x)\). Since \(g(x) = x - 6\), this becomes: \(g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 6\).

Step 3: Find f(g(h(x)))

Finally, substitute \(g(h(x))\) into \(f(x)\). The function \(f(x) = x^4 + 6\) becomes \(f(g(h(x))) = f(\sqrt{x} - 6) = (\sqrt{x} - 6)^4 + 6\).

Step 4: Simplify f(g(h(x)))

To simplify, expand \((\sqrt{x} - 6)^4\). This requires expanding the binomial: \((a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4\). Here, \(a = \sqrt{x}\) and \(b = 6\). Expand to: \((\sqrt{x})^4 - 4(\sqrt{x})^3(6) + 6(\sqrt{x})^2(36) - 4(\sqrt{x})(216) + 6^4\)which simplifies to: \(x^2 - 24x^{3/2} + 216x - 864\). Add back the constant 6, yielding:\(x^2 - 24x^{3/2} + 216x - 858\).

Key Concepts

Understanding Function Composition

Function composition is like stacking multiple functions on top of one another. When you see notation like \(f(g(h(x)))\), it means you apply the innermost function first and then proceed outward. It's analogous to following a recipe, where each function adds a different ingredient to the final dish.
Think of it as a process:

• Start with the inner function \(h(x)\).
• Use its output as the input for the next function, \(g(x)\).
• Finally, use the output from \(g(x)\) as the input for \(f(x)\).

This approach helps in breaking down complex functions into manageable parts. When dealing with multiple functions, remember always to simplify as each step builds upon the previous one, creating a neat chain from input to final output.

The Art of Algebraic Simplification

Simplifying algebraic expressions involves transforming a complex expression into a simpler form without changing its value. In our case, we need to simplify \((\sqrt{x} - 6)^4 + 6\). This often requires careful mathematical steps, like expanding binomials or reducing rational expressions.
Here are some tips for successful simplification:

• Look for common factors that can be factored out.
• Simplify fractions by cancelling out like terms.
• Apply laws of exponents like \(a^m \cdot a^n = a^{m+n}\).

Each step of simplifying reveals the elegance and efficiency of mathematics, showcasing how intricate calculations can turn into tidy results.

Exploring Binomial Expansion

Binomial expansion is a technique for expanding expressions raised to a power, such as \((a-b)^n\). In our example, we need to expand \((\sqrt{x} - 6)^4\), which requires us to apply the binomial theorem. The full expansion results in a series of terms, each involving powers of both \(a\) and \(b\).
Here's how it works for \((a - b)^4\):

• The first term is \(a^4\).
• The second term is \(-4a^3b\).
• Continuing similarly, you derive all the terms up to \(b^4\).

In each term, notice how the powers of \(a\) decrease, while the powers of \(b\) increase, maintaining a comprehensive balance. Binomial expansion simplifies the process of dealing with higher powers by showing patterns that assist in the calculation.

Navigating Nested Functions

Nested functions involve functions within functions, much like layers in a cake. When you evaluate \(f(g(h(x)))\), you're dealing with nested functions, a concept that requires keeping track of each function as it feeds into the next.
To handle nested functions effectively:

• Start by evaluating the innermost function, here \(h(x) = \sqrt{x}\).
• Move outward, using results from one function as the input for the next.
• This keeps computations organized and intercepts errors early on.

Understanding nested functions involves recognizing the order of operations and keeping your mathematical steps aligned, ultimately simplifying the complexity by careful progression.