Problem 85 - College Algebra

Question

Let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((g \circ f)(x)=F(x)\).

Step-by-Step Solution

Answer

False, because \((g \circ f)(x) \neq F(x)\).

Step 1: Identify the Composition Function

The composition function \((g \circ f)(x)\) means we first apply \(f(x)\) and then apply \(g(x)\) to the result of \(f(x)\). In other words, we need to find \(\ g(f(x))\).

Step 2: Evaluate f(x)

Given \(f(x) = x^5\), let's keep this expression as it is since we need it for the next step.

Step 3: Substitute f(x) into g(x)

We have \(g(x) = x + 1\). Substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(x^5) = x^5 + 1\).

Step 4: Compare g(f(x)) with F(x)

We found \(g(f(x)) = x^5 + 1\). Now, compare it to \(F(x) = (x+1)^5\). These expressions are not equivalent.

Step 5: Conclusion

Since \(g(f(x))\) and \(F(x)\) are not the same, \((g \circ f)(x) eq F(x)\). Therefore, the statement is False.

Key Concepts

Algebraic Functions

Algebraic functions are functions that are defined using polynomials. They involve operations such as addition, subtraction, multiplication, division, and roots of expressions. These functions are crucial in mathematics because they describe many real-world phenomena. Let’s delve into each of our functions in this exercise:

• The function \(F(x) = (x+1)^5\) is an example of a polynomial expressed in the form of a power of a binomial. Polynomial operations like this occur frequently in algebra.
• The function \(f(x) = x^5\) is a simpler polynomial, where the output is the fifth power of the input.
• \(g(x) = x + 1\) represents a linear function, adding something (here, 1) to the input.

Understanding algebraic functions is key since they are foundational blocks for more complex expressions like those seen in calculus and other advanced mathematics.

Function Operations

Function operations involve the processes by which two or more functions are manipulated. This could include adding, subtracting, multiplying, dividing, and, importantly for this problem, composing functions.Here’s how each operation works:

• **Addition/Subtraction/Multiplication/Division:** These operations are performed term by term with each corresponding part of the functions.
• **Composition:** A more complex operation, composition involves plugging one function into another. In the exercise, the composition \((g \circ f)(x)\) involves taking the output from \(f(x)\) and using it as the input for \(g(x)\).

In our problem, we first evaluated \(f(x) = x^5\). Then, we substituted this result into \(g(x)\) leading to \(g(f(x)) = x^5 + 1\). This demonstrates that while the concept is straightforward, careful consideration is needed to avoid errors when substituting one function into another.

Mathematical Proof

A mathematical proof is a logical process of showing that a certain statement or theorem is always true, based on acceptable reasoning.In this context, we need to verify whether \((g \circ f)(x) = F(x)\):

• Start by determining each function clearly. For this exercise, we needed to particularly examine \(g(f(x)) = x^5 + 1\) and \(F(x) = (x+1)^5\).
• By comparing the expressions, we observe that \(x^5 + 1\) is different from \((x+1)^5\). The latter expands to \(x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1\), a far more complex expression.

This check proves that the initial statement \((g \circ f)(x) = F(x)\) is false. Mathematical proofs involve showing whether each step logically follows from the previous one, which is why clarity and rigor are essential.