Problem 56 - Precalculus with Limits

Question

In Exercises 53-60, find two functions \(f\) and \(g\) such that \((f \circ g)(x) = h(x)\). (There are many correct answers.) \(h(x) = \sqrt{9-x}\)

Step-by-Step Solution

Answer

Two possible functions such that their composition equals to \(h(x) = \sqrt{9-x}\) are \(f(x) = \sqrt{x}\) and \(g(x) = 9 - x\).

Step 1: Identifying function \(g(x)\)

Function \(g(x)\) will be applied first. We can assume that it is the inside of the function \(h\). The innermost operation in \(h(x)\) is \(9-x\), so we can say \(g(x) = 9 - x\).

Step 2: Identifying function \(f(x)\)

Function \(f(x)\) is applied to the result of \(g(x)\). The outer operation in \(h(x)\) is the square root, so we can write \(f(x) = \sqrt{x}\).

Step 3: Checking the found functions

To make sure these functions are correct, we can do the composition of \(f\) and \(g\) and check if we obtain \(h(x)\). \(f(g(x)) = f(9-x) = \sqrt{9-x}\) which is equal to \(h(x)\).

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental concept in mathematics. It involves finding a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. When dealing with functions, the square root operation is typically represented as \( f(x) = \sqrt{x} \). This function only accepts non-negative values as input, since the square root of a negative number is not defined within the real number system.

In our case, the function \( h(x) = \sqrt{9-x} \) uses the square root to process the expression inside it, \( 9-x \). The result of this operation must be non-negative, which implies that \( 9-x \geq 0 \). Solving this inequality shows that \(x\) must be less than or equal to 9 for the function to have real output.

Exploring the Difference of Terms

The "difference of terms" refers to operations that result in subtraction, which is a key arithmetic operation. In our given function \( h(x) = \sqrt{9-x} \), the phrase "difference of terms" directly relates to the expression \( 9-x \). Here, 9 is the minued and \( x \) is the subtrahend. This type of expression is common in algebra, and it often appears within more complex functions to adjust how the variable is processed.

Understanding \( 9-x \) involves recognizing it as an operation that reverses the value of \( x \) concerning 9. As \( x \) increases, \( 9-x \) decreases, which can affect other operations in composite functions. Comprehending how differences work in math allows us to predict the behavior of functions easily, especially when combined with other operations like square roots.

Demystifying Function Operations

Function operations include various ways of combining functions to create new ones. The primary operations are addition, subtraction, multiplication, division, and composition. In our exercise, we focus particularly on function composition. Composition involves applying one function to the results of another function, and it's often represented as \((f \circ g)(x) = f(g(x))\). This operation allows us to build complex expressions from simpler components.

In the example given, we find separate functions \( f(x) = \sqrt{x} \) and \( g(x) = 9-x \) whose composition gives the original \( h(x)\). Here's how it works:

• First, apply \( g(x) = 9-x \) to an input \( x \).
• Next, take the result \( g(x) \) and use it as the input for \( f(x) = \sqrt{x} \).
• This results in \( f(g(x)) = \sqrt{9-x} \), which matches \( h(x) \).

Function composition simplifies problem solving by breaking down complex problems into manageable steps. Understanding this concept is crucial for working with multiple-layered mathematical functions.