Problem 65 Find \(f(x)\) and \(g(x)\) such ... [FREE SOLUTION] (2024)

Expert verified

Function composition is an important concept in mathematics. It involves putting one function inside another to produce a new function. If we have two functions, say\( f(x) \) and \( g(x) \), the composition of these functions is written as \( (f \, \text{∘} \, g)(x) \), which is read as 'f of g of x'. This means we first apply \( g(x) \) and then apply \( f \) to the result of \( g(x) \). In simpler terms, it is like doing two operations in sequence:

• First, apply \( g(x) \),
• Then, apply \( f \) to the result of \( g(x) \).

To break it down further, if \( g(x) = x^3 \) and \( f(x) = \frac{x + 1}{x - 1} \), then the composition \( (f \, \text{∘} \, g)(x) \) becomes \( f(g(x)) \). This can be understood as:

• Substitute \( x \) with \( x^3 \) in the outer function \( f \):


• \( f(x^3) = \frac{x^3 + 1}{x^3 - 1} \).

This process helps in simplifying complex expressions by breaking them down into manageable parts.

The inner function is the function that is first applied in the composition process. It's like the first step in a cooking recipe. In the example problem, \r h(x) = \frac{x^3 + 1}{x^3 - 1}, the inner function \( g(x) \) is chosen to simplify the expression applied later by the outer function. By selecting \( g(x) = x^3 \), we set ourselves up for easier manipulation.
Here's the thought process:

• Look at \( h(x) \), which is \( \frac{x^3 + 1}{x^3 - 1} \).
• Notice a common structure in both the numerator and denominator, \( x^3 \).
• Choosing \( g(x) = x^3 \) simplifies the inner part of the expression, making further steps more straightforward.

The right choice of the inner function can make or break a problem, as it lays the foundation for applying the next function effectively.

The outer function is applied to the result of the inner function. It's like the final touch in a recipe. After we’ve performed the inner function \( g(x) \), we apply the outer function \( f(x) \).

For the given problem, we already know our inner function is \( g(x) = x^3 \). Now, we determine the outer function \( f(x) \):

• Replace \( x^3 \) in the original equation: \( h(x) = \frac{x^3 + 1}{x^3 - 1} \) becomes
• \( f(x^3) = \frac{x^3 + 1}{x^3 - 1} \).
• Now, set \( z = x^3 \), then \( f(z) = \frac{z + 1}{z - 1} \).
• Hence, our outer function is \( f(x) = \frac{x + 1}{x - 1} \).

The outer function finalizes the composition, turning the results of the inner function into the final answer. By carefully choosing it, we ensure that the original composite function \( h(x) \) is accurately represented.

Verification is the last but crucial step to ensure that our selected functions \( f(x) \) and \( g(x) \) work correctly together to form \( h(x) \). For this example:

\r

• We need to check that \( f(g(x)) = h(x) \).
• Insert \( g(x) = x^3 \) into \( f: f(x^3) = \frac{x^3 + 1}{x^3 - 1} \).
• Since this matches \( h(x) = \frac{x^3 + 1}{x^3 - 1}\), our functions are verified.

Verification confirms:

• The inner function \( g(x) \) and
• The outer function \( f(x) \)

are correctly identified. This step ensures confidence in our solution, confirming all parts fit together as intended.