Problem 38 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\left|x^{2}+7\right| $$

Step-by-Step Solution

Answer

Choose \(g(x) = x^2 + 7\) and \(f(x) = |x|\).

Step 1: Understand the Problem Statement

We need to express the given function \( h(x) = |x^2 + 7| \) in the composite form \( h(x) = f(g(x)) \). This means identifying a function \( g(x) \) that will be processed first and then subsequently modified by the function \( f(x) \).

Step 2: Identify an Inner Function

For the function \( h(x) = |x^2 + 7| \), the expression inside the absolute value, \( x^2 + 7 \), is a natural choice for the inner function \( g(x) \). Therefore, we define \( g(x) = x^2 + 7 \).

Step 3: Determine the Outer Function

With \( g(x) = x^2 + 7 \), we can replace this inside the \( h(x) \), leading to \( h(x) = |g(x)| \). Hence, the outer function \( f(x) \) is simply the absolute value function, \( f(x) = |x| \).

Step 4: Verify the Composite Function

To ensure that \( f(g(x)) = h(x) \), substitute \( g(x) \) into \( f(x) \). We have \( f(g(x)) = f(x^2 + 7) = |x^2 + 7| \), which matches the original function \( h(x) \). This confirms our choices for \( f(x) \) and \( g(x) \).

Key Concepts

Absolute value function

The absolute value function is a special kind of function that transforms any input into its non-negative equivalent. This function is usually denoted by vertical lines, like \(|x|\).
It is defined as follows:

• \(f(x) = x\) if \(x \geq 0\)
• \(f(x) = -x\) if \(x < 0\)

Delivering the positive magnitude of any given number, the absolute value function is crucial in situations where direction or sign needs to be ignored.
As in the current exercise, it handles the expression \(x^2+7\) by returning the non-negative value regardless of the input's sign. This behavior of wrapping an expression inside an absolute value makes it a perfect candidate for an outer function in composite function scenarios.

Function decomposition

Function decomposition involves breaking down a complex function into simpler, smaller functions. In the given exercise, we are tasked with splitting the function \( h(x) = |x^2 + 7| \) into two individual functions \(f(x)\) and \(g(x)\).
This approach helps in simplifying problems and making complex functions easier to manipulate, understand, and analyze.
By identifying \(g(x) = x^2 + 7\) and \(f(x) = |x|\), we have successfully decomposed \(h(x)\) into simpler components. This strategy is valuable in mathematics for dealing with advanced concepts like derivatives, integrals, and more complex transformations.

Inner function

The inner function is the part of a composite function that acts as the initial transformation of the input value. In our exercise, \(g(x) = x^2 + 7\) is the inner function.
It transforms any input \(x\) into \(x^2 + 7\). This simplified expression is then passed to the outer function for further transformation.
Choosing the right inner function typically involves identifying a natural grouping or simplification in the original expression. The inner function often sets the stage for the process that follows, making it a cornerstone of function decomposition.

Outer function

The outer function is the second function that acts on the result of the inner function within a composite function setup. In the exercise at hand, \(f(x) = |x|\) serves as the outer function.
This function takes the result from the inner function \(g(x) = x^2 + 7\) and applies the absolute value operator to it, ensuring every output is non-negative.
The outer function is majorly responsible for the final transformation that matches the original function's requirement. Selecting \(f(x) = |x|\) was essential here as it matches \(h(x)\)'s specification, confirming that the step-by-step breakdown was correctly executed.