Problem 29 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=\frac{4}{(x+2)^{2}} $$

Step-by-Step Solution

Answer

Functions are \(f(x) = \frac{4}{x^2}\) and \(g(x) = x + 2\).

Step 1: Identify Inner Function

To find the functions \(f(x)\) and \(g(x)\), let's start with identifying the inner function \(g(x)\). Looking at the structure \(h(x) = \frac{4}{(x+2)^2}\), we notice that the expression \((x+2)\) is a transformation applied to \(x\). Therefore, we can choose \(g(x) = x + 2\).

Step 2: Identify Outer Function

Now, with \(g(x) = x + 2\), we need to reformulate \(h(x)\) in terms of \(g(x)\). Substitute \(x + 2\) with \(g(x)\), yielding \(h(x) = \frac{4}{g(x)^2}\). This means the outer function \(f(x)\) can be expressed as \(f(x) = \frac{4}{x^2}\).

Step 3: Verify Composite Function

Finally, verify that \(h(x) = f(g(x))\) holds true. Substitute \(g(x) = x + 2\) into the function \(f(x) = \frac{4}{x^2}\) to get: \(f(g(x)) = \frac{4}{(x+2)^2}\), which matches the original \(h(x)\). Thus, \(h(x) = f(g(x)) = \frac{4}{(x+2)^2}\) is confirmed.

Key Concepts

Inner Function

When dealing with function composition, the first step is identifying the **inner function**. In mathematical terms, the inner function is the expression inside another function. Look at the given function, \(h(x) = \frac{4}{(x+2)^2}\). Here, we notice that

• \(x + 2\) is the original variable that undergoes transformation first.

Thus, we define our inner function as \(g(x) = x + 2\). This transformation simplifies the elements within the larger function, preparing it for further operations.
Finding the inner function is essential because it sets the stage for identifying the outer function. Without it, composing the function into simpler parts would be difficult, making the entire process complex.

Outer Function

Once we have determined the inner function, the next step is to identify the **outer function**. This is the function that acts upon the inner function. In our scenario, we have the complete expression

• \(h(x) = \frac{4}{g(x)^2}\),

where \(g(x) = x + 2\).
To find the outer function, observe what operation is performed on \(g(x)\). We see that it is squared and then placed in the denominator of a fraction. Therefore, our outer function can be expressed as \(f(x) = \frac{4}{x^2}\).
This format shows how the outer function acts on whatever is inside it, which is the result from the inner function. It's important to see how the outer function shapes the final composition by guiding the transformation on \(g(x)\).

Composite Function

Combining the inner and outer functions gives us the **composite function**. This demonstrates how two functions work together seamlessly to form a whole. For the given problem, we have established that

• \(g(x) = x + 2\),
• and \(f(x) = \frac{4}{x^2}\).

To verify the composite function, compute \(f(g(x))\). Substitute \(g(x)\) into \(f(x)\), resulting in \(f(g(x)) = \frac{4}{(x+2)^2}\). This matches exactly with \(h(x)\) and confirms that our function composition is correct.
A composite function takes the output from the inner function and feeds it into the outer function, integrating their operations. Understanding this concept not only simplifies complex expressions but also deepens comprehension of how functions interact in mathematics.