Problem 26 - Algebra A Combined Function

Question

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ G(x)=\sqrt{3 x} $$

Step-by-Step Solution

Answer

The function can be written as the composition \( G(x) = g(f(x)) \).

Step 1: Identify the Composition Format

We are given the function \( G(x) = \sqrt{3x} \). Let's identify how this function can be expressed as a composition of the given functions, \(f(x)=3x\), \(g(x)=\sqrt{x}\), and \(h(x)=x^2+2\).

Step 2: Determine the Outer Function

The outermost operation in \( G(x) = \sqrt{3x} \) is the square root. Thus, the outer function is \( g(x) = \sqrt{x} \).

Step 3: Determine the Inner Function

The expression inside the square root of \( G(x) = \sqrt{3x} \) is \(3x\). This matches the function \(f(x) = 3x\).

Step 4: Write the Composition

Since \( f(x) = 3x \) is inside the square root, and \( g(x) = \sqrt{x} \) is applied to this result, we can express \( G(x) \) as a composition of \(g\) and \(f\): \( G(x) = g(f(x)) \).

Key Concepts

Composition of Functions

In mathematics, the composition of functions is a critical concept used to combine two or more functions to form a new function. It's like stacking two operations on top of each other in sequence. If you have two functions, say \( f \) and \( g \), you can create a new function, denoted by \( g(f(x)) \) or \( (g \circ f)(x) \).

Function composition is all about applying one function to the results of another. So basically, you take the output of one function and use it as the input to another one. This is a helpful technique, especially in analyzing complex problems by breaking them into more manageable parts.

For instance, in the problem provided, the function \( G(x) = \sqrt{3x} \) is expressed as a composition of \( g(x) = \sqrt{x} \) and \( f(x) = 3x \). This means \( G(x) \) becomes \( g(f(x)) \), which translates to applying \( g \) to the result of \( f \).

Outer Function

The outer function in a composition is the one that is applied last in sequence. For the given problem, when expressing \( G(x) = \sqrt{3x} \) as a composition, we need to determine which function acts as the outermost wrapper.

In this case, the squaring operation \( g(x) = \sqrt{x} \) serves as the outer function because it wraps around the inside computation of \( f(x) = 3x \). Think of it this way: the outer function dictates the overall output shape or form of the resultant function.

Finding the outer function is central to understanding how each function in the composition impacts the overall transformation of the input values.

Inner Function

The inner function is the first function applied to the input in a composition of functions. It's the function that's nestled inside the outer function. In our task, when we look at \( G(x) = \sqrt{3x} \), the inner function is \( f(x) = 3x \).

Here's how it works: initially, \( f \) takes the input \( x \) and multiplies it by 3. Whatever result comes from \( f(x) \) then becomes the input for the outer function \( g(x) = \sqrt{x} \).

Understanding the role of the inner function helps clarify the starting point of the operations in a composite structure. It's the initial transformation that sets the stage for the subsequent outer function to complete the function composition.

• The inner function handles the starting transformation of \( x \).
• It dictates the flow of values that the outer function will shape.