Problem 38 - College Algebra

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(\mathrm{g}(x))\) $$h(x)=\left|x^{2}+7\right|$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Inner Function

We want to express the function \( h(x) = \left| x^2 + 7 \right| \) in the form of \( h(x) = f(g(x)) \). First, we identify \( g(x) \) such that any operations inside the absolute value function are handled by \( g(x) \). So, we let \( g(x) = x^2 + 7 \).

Step 2: Define the Outer Function

Next, we need to determine what function \( f(x) \) will transform \( g(x) \) to produce \( h(x) \). Since \( h(x) = \left| x^2 + 7 \right| \), we know that \( f(x) \) has to be the absolute value function to complete the transformation. Therefore, we have \( f(x) = |x| \).

Step 3: Conclude the Function Decomposition

So, we have expressed the function \( h(x) = \left| x^2 + 7 \right| \) as \( h(x) = f(g(x)) \), where \( f(x) = |x| \) and \( g(x) = x^2 + 7 \).

Key Concepts

Composite Functions

Composite functions are formed when one function is applied to the result of another function. This means the output of the first function becomes the input of the second. In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), their composite can be written as \( f(g(x)) \). This is read as "\( f \) of \( g \) of \( x \)".

• Composite functions allow you to combine multiple operations into a single expression.
• They are particularly useful in simplifying complex mathematical models.
• The key is understanding the order of operations: you always apply the inside function first.

In our exercise, the function is expressed as \( h(x) = f(g(x)) \) where \( g(x) = x^2 + 7 \) and \( f(x) = |x| \). Here, you apply \( g(x) \) first to handle any algebraic operations before applying the absolute value operation of \( f(x) \). This allows the problem to be broken down into manageable parts, making it easier to analyze and understand.

Absolute Value Function

The absolute value function, denoted \( |x| \), plays an essential role in mathematics due to its ability to convert any real number into a non-negative value. It is defined as:

• If \( x \ge 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

This function is particularly useful when dealing with real-world problems requiring positive quantities or when analyzing distance.
If you consider the exercise, the absolute value function \( f(x) = |x| \) is applied after calculating \( x^2 + 7 \). This ensures that the entire composite function \( h(x) = \left| x^2 + 7 \right| \) is non-negative, regardless of the value of \( x \). By wrapping this expression within absolute value brackets, you effectively limit the range of \( h(x) \) to all non-negative real numbers.

Inner and Outer Functions

In the context of composite functions, the concepts of inner and outer functions are crucial.

• The inner function is the function that you first apply, before applying the outer function. It handles the initial transformation of the input variable.
• The outer function is applied to the result of the inner function to give the final output of the composite function.

For our exercise, the inner function is \( g(x) = x^2 + 7 \). This inner function squares the input and adds 7, setting the stage for the next transformation by the outer function.
The outer function is \( f(x) = |x| \), which processes the result from \( g(x) \) by taking its absolute value, ensuring the final result is always non-negative. Understanding which function serves as the inner and which as the outer helps break down complex expressions into a series of simple calculations, giving clarity and an organized pathway to solving function-based problems.