Problem 85 - College Algebra

Question

For the following exercises, 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; \((g \circ f)(x) = x^5 + 1\), which is not equal to \(F(x)=(x+1)^5\).

Step 1: Understand the Composition of Functions

The notation \((g \circ f)(x)\) represents the composition of the function \(g\) with \(f\), meaning \(g(f(x))\). We'll substitute \(f(x)=x^5\) into \(g(x)=x+1\).

Step 2: Substitute f(x) into g(x)

Substitute \(f(x)=x^5\) into \(g(x)\). This gives us the expression \(g(f(x))=g(x^5)\). Since \(g(x)=x+1\), we replace \(x\) with \(x^5\) to get \(x^5+1\).

Step 3: Compare with F(x)

Now, compare the expression \(g(f(x))=x^5+1\) obtained from Step 2 with \(F(x)=(x+1)^5\). Analyze whether both expressions are the same or different.

Step 4: Conclusion

Since \(g(f(x))=x^5+1\) is not equal to \(F(x)=(x+1)^5\), it is evident that the statement \((g \circ f)(x)=F(x)\) is False.

Key Concepts

Function Notation

Function notation is a way to express the behaviors and operations of functions using symbols and expressions. Instead of saying "the function of x," we use a simpler notation: \(f(x)\). This form tells us that \(f\) is a function depending on the variable \(x\).

Function notation doesn't just make mathematics more concise; it also allows us to easily express operations involving functions, such as transformations, combinations, or compositions. When you see notations like \((g \circ f)(x)\), this is telling you that function \(f\) is applied to \(x\) first, and then function \(g\) is applied to the result of \(f(x)\).

Using function notation is crucial for exploring deeper mathematical concepts and advanced mathematical problems, making it an essential tool in mathematics.

Polynomials

Polynomials are algebraic expressions made up of terms, which consist of variables raised to whole number exponents and coefficients. A simple example is \(x^2 + 3x + 2\), where each term is part of the polynomial.

The degree of a polynomial is determined by the highest exponent of the variable in the expression. For instance, in the polynomial \(x^5\), the degree is 5. This is important since it gives the polynomial certain characteristics, such as the number of roots it can have and the general shape of its graph.

Polynomials can be added, subtracted, multiplied, and even composed with other functions. These operations enable us to maintain control over complex algebraic expressions. Understanding polynomials is fundamental to succeeding in higher levels of algebra and calculus, where they frequently appear.

Function Substitution

Function substitution is a process where we replace a variable in a function with another expression or function. This is often used in the context of composition, as seen in function composition exercises.

In our example, we have two functions: \(f(x)=x^5\) and \(g(x)=x+1\). To substitute \(f\) into \(g\), we replace every occurrence of \(x\) in \(g(x)\) with \(f(x)\). Therefore, substituting \(f(x)\) into \(g(x)\) results in \(g(f(x)) = g(x^5) = x^5 + 1\).

This process is instrumental in solving complex equations and understanding how different functions interact. Grasping the concept of substitution helps to simplify and solve equations that might initially seem daunting, by breaking them down into more manageable parts.