Problem 34 - Algebra A Combined Function

Question

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=(3 x+4)^{2}+3 $$

Step-by-Step Solution

Answer

\(f(x) = x^2 + 3\) and \(g(x) = 3x + 4\).

Step 1: Identify the Composite Function

The function given is \(h(x) = (3x+4)^2 + 3\). We need to decompose \(h(x)\) into two functions, \(f(x)\) and \(g(x)\), such that \(h(x) = (f \circ g)(x) = f(g(x))\).

Step 2: Define Inner Function

To start decomposing, let's identify \(g(x)\), the inner function, which can be the expression inside the square in \((3x+4)^2\). Define \(g(x) = 3x + 4\).

Step 3: Define Outer Function

Next, determine \(f(x)\), the outer function. It should include applying the square to the result of \(g(x)\) and then adding 3. So, if \(u = g(x) = 3x + 4\), then \(f(u) = u^2 + 3\).

Step 4: Verify the Decomposition

Plug \(g(x) = 3x + 4\) into \(f(x)\) to verify: \(f(g(x)) = ((3x + 4)^2 + 3) = h(x)\). This confirms the correct decomposition of \(h(x)\) into \(f(x)\) and \(g(x)\).

Key Concepts

Composite Functions

When you hear about composite functions, think about them as functions within functions. It’s a layered approach. Imagine placing a smaller box into a larger one. In mathematics, the smaller box is the inner function, and the outer box is the outer function. When combined, they create the composite function.

To grasp this with an example, consider the function notation \(f \circ g\). This reads as "f composed with g". Here, the value that results from the inner function \(g(x)\) is placed into the outer function \(f(x)\), effectively nesting the functions together. Recognizing this concept is essential in function decomposition, where the task is to determine the specific roles of each function.

Inner Function

The inner function is the initial step in our composing journey. It acts as the base layer upon which the outer function builds its operations. In the problem given, we have the function \(h(x) = (3x+4)^2 + 3\). Our goal is to find \(g(x)\) or the inner function.

To identify \(g(x)\), we look for the expression directly used within the main compound operations—in this case, the expression within the parentheses. It represents the initial transformation of our input \(x\).

• In \(h(x)\), we notice the expression \(3x + 4\) inside the squared term.
• Thus, we define the inner function as \(g(x) = 3x + 4\).

The inner function’s role is pivotal as it determines the output that becomes the input for the outer function.

Outer Function

Once we have the inner function, it's time to determine \(f(x)\), often referred to as the outer function. This function builds on the output of \(g(x)\).

In our exercise, after establishing that \(u = g(x) = 3x + 4\), the outer function needs to manipulate this result further. We look at the remaining operations performed on \(u\). Here they are:

• The outer operation takes \(u\), squares it, and adds 3.
• Therefore, the outer function is defined as \(f(u) = u^2 + 3\).

Recognizing \(f(x)\) is key because it completes the transformation your input undergoes in forming the composite function \(h(x)\).

Function Verification

The last step in validating our decomposition involves function verification. It’s akin to a mathematical quality control. We ensure all assembled parts work together correctly.

To verify, substitute \(g(x) = 3x + 4\) into \(f(g(x))\) and check whether it equates to \(h(x) = (3x+4)^2 + 3\).

• Plugging in: \(f(g(x)) = f(3x + 4)\).
• This gives us: \((3x+4)^2 + 3\).

This outcome matches the original \(h(x)\), confirming our split into \(f(x)\) and \(g(x)\) is accurate. Verification assures your mathematical solution is as solid as a carefully constructed puzzle, ensuring a seamless fit of all its pieces.