Problem 49 - Elementary and Intermediate Algebra

Question

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$ h(x)=(x+15)^{2} $$

Step-by-Step Solution

Answer

Choose \(f(x) = x^2\) and \(g(x) = x + 15\) to get \((f \circ g)(x) = (x + 15)^2\).

Step 1: Understand Composition of Functions

The composition \((f \circ g)(x)\) means \(f(g(x))\). We need to find two functions \(f(x)\) and \(g(x)\) such that when \(g(x)\) is plugged into \(f(x)\), it results in \(h(x) = (x + 15)^2\).

Step 2: Choose an Inner Function \(g(x)\)

To find \(g(x)\), notice that the inside of the square \((x + 15)\) suggests a natural choice for \(g(x)\). Let \(g(x) = x + 15\). This function represents the modification of \(x\) before squaring.

Step 3: Determine the Outer Function \(f(x)\)

Given that \(g(x) = x + 15\), the expression \((x + 15)^2\) suggests that \(f(x)\) should be squaring the input. So, define \(f(x) = x^2\).

Step 4: Verify the Composition

Verify that substituting \(g(x)\) into \(f(x)\) gives \(h(x)\). Compute \(f(g(x)) = f(x + 15) = (x + 15)^2\), which matches \(h(x)\). Therefore, \(f(x) = x^2\) and \(g(x) = x + 15\) work as the functions such that \(h(x) = (f \circ g)(x)\).

Key Concepts

Inner Function

When dealing with function composition, identifying the **inner function** is a key step. This function is the initial process applied to our input variable, often making an obvious transition of the input before further operations are applied.
In our exercise, the expression \(x + 15\) serves as the part of the function suggesting change right before squaring occurs. Therefore, it makes sense to define it as our inner function \(g(x)\).

• **Simple definition:** Consider what happens first to the input.
• **In context:** Here, \(g(x) = x + 15\) represents a simple shift operation applying to every input \(x\).

Recognizing \(g(x)\) is just about pinpointing the initial adjustment in any function composition situation.

Outer Function

After identifying the inner function, it's time to establish the **outer function**. This function usually completes the transformation begun by the inner function, finalizing the composition.
For the given problem where \(h(x) = (x + 15)^2\), the task is to square the result of \(g(x)\). Thus, the outer function \(f(x)\) would naturally be chosen as squaring whatever input it receives, leading us to define \(f(x) = x^2\).

• **Role of outer function:** Finalizes the composite function.
• **In context:** Further modifies the changed input from \(g(x)\). In this case, it squares \(x + 15\).

Defining the outer function solidifies the purpose and transformation that \(h(x)\) intends to achieve.

Verifying Compositions

Once both the inner and outer functions are identified, it's crucial to **verify that the composition** results in the original function, \(h(x)\). Verification ensures the definitions of \(f(x)\) and \(g(x)\) are correct and functioning as intended.
By substituting \(g(x)\) into \(f(x)\), we find:\[ f(g(x)) = f(x + 15) = (x + 15)^2 \]
This results in \(h(x)\), satisfying our initial equation. This step assures us that the functions were correctly determined.

• **Checking correctness:** Substitute the inner function into the outer function.
• **Result alignment:** Ensure that the result matches \(h(x)\).

Verification is a final assurance that the setup and operations are done correctly.