Problem 46 - Algebra and Trigonometry

Question

\(45-50\) Express the function in the form \(f \circ g\) $$ F(x)=\sqrt{x}+1 $$

Step-by-Step Solution

Answer

The functions are \(f(u) = \sqrt{u} + 1\) and \(g(x) = x\).

Step 1: Identify Inner and Outer Functions

To express the function in the form \(f \circ g\), we need to identify two functions where one function is the inner function \(g(x)\) and the other, \(f(x)\) is the outer function. The composition \(F(x) = f(g(x))\).

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

Let's identify the innermost operation in the function \(F(x) = \sqrt{x} + 1\). We start with \(g(x) = x\). However, to represent the part inside the square root as an entire expression, let's instead choose \(g(x) = x\).

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

Since the expression inside \(F(x)\) is \(\sqrt{g(x)} + 1\), we will create the function \(f(u)\) as \(\sqrt{u} + 1\) where \(u = g(x)\).

Step 4: Write the Function Composition

Now, combine \(g(x)\) and \(f(x)\) into the composition. We have \(g(x) = x\), hence \(f(g(x)) = f(x) = \sqrt{x} + 1\). This matches \(F(x)\).

Step 5: Verify the Composition

Verify that \(f \circ g = F\): since we defined \(f(g(x)) = \sqrt{x} + 1\) and this matches our original function, the decomposition is verified.

Key Concepts

Inner Function

When dealing with function composition, **the inner function** is a crucial element. It is the function that is inside another function. Think of it as the core operation that your main function must first execute. This function is essential for forming the basis of a composition.

In the example of the function composition for \(F(x) = \sqrt{x} + 1\), you need to choose the inner function carefully. The primary inner operation we can spot is under the square root, suggesting that \(g(x) = x\) can effectively function as the inner expression. Here, \(g(x)\) may seem trivial, but correctly identifying it ensures your composition fits perfectly.

Some key points about the inner function include:

• It forms the input of the outer function.
• Identifying it requires examining the nested parts of the expression.
• In the composition \(f(g(x))\), \(g(x)\) represents the core value that is transformed by \(f(x)\).

Outer Function

Understanding the outer function in function composition is just as crucial as knowing the inner function. **The outer function** operates on the result of the inner function, enveloping and transforming it further. It is the final act in the sequence that delivers the full result.

For instance, in composing the function \(F(x) = \sqrt{x} + 1\), our outer function can be expressed as \(f(u) = \sqrt{u} + 1\). Here, \(u\) represents the input received from the inner function \(g(x)\). This clearly shows that the outer function builds on the results of the inner function, adding a step to the end of the computation.

Consider these important points about the outer function:

• It utilizes the output of the inner function.
• Defining it usually involves substituting the result of the inner function into its formula.
• It frames the larger picture and completes the function composition process.

Square Root Function

The **square root function** is a distinctive function that can often appear as part of a composition in mathematics. This function, denoted typically as \(\sqrt{x}\), handles numbers that need to be expressed in terms of their square root, greatly impacting the understanding of function compositions.

In our example, the primary challenge is to see how the square root function plays into the larger function such as \(F(x) = \sqrt{x} + 1\). Here, the square root is central to the system, making up a key part of both the inner and outer functions in the composition.

Some essential aspects of the square root function include:

• It reduces a number to its radical equivalent, impacting the hierarchy in operations.
• In compositions, it often informs where the boundary between inner and outer functions might lie.
• Understanding its role helps in correctly evaluating composite functions without errors.

This analytical view allows for easier problem-solving and deciphering various mathematical scenarios where the square root function is a dominant component.