Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

The two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) are \(f(x) = \sqrt{x}\) and \(g(x) = 9 - x\).

Step 1: Identifying the inner function

Looking at the original function \(h(x) = \sqrt{9-x}\), we can identify \(9-x\) as the inside function. Thus we set \(g(x) = 9 - x\).

Step 2: Identifying the outer function

The rest of the original function that remains after establishing \(g(x)\) is \(\sqrt{y}\), where \(y\) is the outcome of the inner function, this can be identified as the outer function. We thus assign this function to \(f\) and set \(f(x) = \sqrt{x}\).

Step 3: Checking the solution

To verify our approach, the functions \(f\) and \(g\) should compose to form the original function \(h(x)\). This means that \(f(g(x)) = h(x)\). Substituting \(f\) and \(g\) yields \(\sqrt{9 - x} = \sqrt{9 - x}\), which shows that the chosen inverses are correct.

Key Concepts

Functions

Functions are a fundamental concept in mathematics, especially for dealing with various real world problems.
The core idea of a function is to associate an input with exactly one output.
Here's how they work:

• A function takes an input known as an 'independent variable' and applies a specific rule to produce an output.
• The rule could be anything, such as adding a number, taking a square root, or any kind of mathematical operation.

Functions are represented as notations where you typically see expressions like \( f(x) \), which means that function \( f \) takes input \( x \).
This notation simplifies the process of working with mathematical expressions and ensures clarity.

Function Notation

Function notation is the second concept that is essential to understand how we symbolically represent functions.
It provides a clear and concise way to write equations, which makes analyzing and solving them straightforward.
Here’s what you should know:

• Function notation uses letters like \( f \), \( g \), or \( h \) to represent the function's name and \( x \) as the variable.
• When we see \( f(x) = y \), it tells us that when \( x \) is put into function \( f \), it yields output \( y \).
• This notation is flexible, allowing us to easily designate different functions with different names and simplify complex operations like compositions of functions.

When working with complex functions, you can also break them down into smaller parts using function notation, making it possible to perform calculations systematically and accurately.

Square Root Function

The square root function is one of the most fundamental and recognizable types of functions in mathematics.
Characterized by its distinct symbol \( \sqrt{\cdot} \), it plays an essential role in various fields.

• The square root function, written as \( f(x) = \sqrt{x} \), takes a number \( x \) and returns its square root.
• This means for any non-negative number \( x \), \( f(x) \) finds the number which, when multiplied by itself, gives \( x \).

The square root function is particularly interesting because:

• It is defined only for non-negative numbers, as square roots of negative numbers are not real numbers.
• It can result in numbers that are not integers, a characteristic that emphasizes the need for a deep understanding of mathematical numbers.
• It often appears in solving quadratic equations and is crucial for understanding geometry, among other disciplines.

Understanding how and when to use the square root function is critical for students as it is a stepping stone to more advanced mathematical concepts.