Problem 32 - College Algebra

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(\mathrm{g}(x))\) $$h(x)=\frac{1}{\left(3 x^{2}-4\right)^{-3}}$$

Step-by-Step Solution

Answer

The functions are \(f(x) = x^3\) and \(g(x) = 3x^2 - 4\).

Step 1: Identify the Inner Function

To express the function given, identify the inner function, typically an expression inside another function. In this case, consider the expression inside the parenthesis of the power, which is \(g(x) = 3x^2 - 4\).

Step 2: Identify the Outer Function

Next, identify the outer function \(f(x)\). For our function, the expression can be rewritten as \((g(x))^{-3}\). Hence, \(f(x)\) must be a function that applies to \(g(x)\). Thus, \(f(x) = x^{-3}\). However, since the problem also includes inversion of this term, adjust \(f(x)\) accordingly: \(f(x) = \frac{1}{x^{-3}} = x^3\).

Step 3: Write the Composite Function

Combine the two identified functions to form \(h(x)\). Thus, the function \(h(x)\) is given by \(h(x) = f(g(x))\), which results in \(h(x) = \left((3x^2 - 4)^{-3}\right)^{-1}\), aligning with \(h(x) = (g(x))^3\).

Step 4: Verify the Composite Function

To ensure correctness, substitute \(g(x)\) into \(f(x)\) to check if it equals \(h(x)\). For \(f(x) = x^3\), substituting \(g(x) = 3x^2 - 4\) results in \(f(g(x)) = (3x^2 - 4)^3\), which simplifies to \(h(x)\). Our step confirms alignment with the original function \(h(x) = \frac{1}{\left(3x^2 - 4\right)^{-3}}\).

Key Concepts

Composite Functions

A composite function is like a function within another function. You create a composite function when you apply one function to the result of another function. It's written as \( f(g(x)) \) or sometimes simply \( (f\circ g)(x) \), meaning you're applying \( g(x) \) first, and then \( f \) to the outcome of \( g(x) \).
This concept is really helpful in breaking down complex problems into simpler steps. When dealing with composite functions, you're effectively nesting two functions, like putting one box inside another. Start by identifying which small operations happen first (inner function), and then how those results are manipulated (outer function).
Understanding composite functions means mastering two main ideas:

• The order of operations: always apply the inner function before the outer one.
• Combining results: substitute the expression for \( g(x) \) directly into \( f(x) \).

By turning a complex function into simpler composite parts, you can often solve challenging exercises more easily.

Inner and Outer Functions

When dealing with decomposition of functions, two terms pop up frequently: inner and outer functions. Think of the inner function, \( g(x) \), as the innermost operation within a composite function that you do first. For instance, in the exercise where \( h(x) = \frac{1}{(3x^2-4)^{-3}} \), the inner function is \( g(x) = 3x^2-4 \).
Once you've handled \( g(x) \), move on to the outer function, \( f(x) \), which encompasses the inner function. Here, it means applying the operation to the result of \( g(x) \). In the mentioned exercise, the outer function is initially interpreted as \( f(x) = x^{-3} \) before adjusting to \( f(x) = \left(\frac{1}{x^{-3}} = x^3\right) \).
Here’s how to identify these elements:

• Inner Function: Dig into the formula and locate the expression or component executed first within the parentheses.
• Outer Function: Recognize the operation applied to whatever results from the inner function. It wraps around the inner function and finalizes the calculation.

Understanding these two roles allows one to leverage the power of function decomposition effectively. It simplifies solving more complex expressions by focusing on small, manageable parts.

Algebraic Manipulation

Algebraic manipulation is a set of strategies used to transform and simplify expressions so they can be more easily understood or solved. It often involves rewriting expressions in a different form but keeping their value unchanged.
In the exercise where \( h(x) = \frac{1}{(3x^2-4)^{-3}} \), algebraic manipulation helps by simplifying the expression to locate the appropriate inner and outer functions. Decompose by separating the power and inversion elements, which lets you reframe \( h(x) \) as a composite functioning involving simpler parts of \( f(x) \) and \( g(x) \).
Key steps in effective algebraic manipulation include:

• Factorizing polynomials or expressions when needed.
• Applying basic algebraic rules, like distributive or associative properties, to simplify parts of the expression.
• Isolating terms to clearly define functions \( f \) and \( g \).
• Reversing operations, as in the case of inverting the power in the initial exercise solution, to further simplify.

By mastering algebraic manipulation, you can decode even the trickiest functions and spot the hidden relationships between components. This in turn elevates your problem-solving skills in mathematics.