Problem 86 - College Algebra

Question

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

Step-by-Step Solution

Answer

True, \((f \circ g)(x) = F(x)\).

Step 1: Understanding the Composition

The function composition \((f \circ g)(x)\) means that we are substituting \(g(x)\) into \(f(x)\). So, \((f \circ g)(x) = f(g(x))\).

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

Given \(g(x) = x+1\) and \(f(x) = x^5\), substitute \(g(x)\) into \(f(x)\). This gives us \(f(g(x)) = (x+1)^5\).

Step 3: Comparing with F(x)

Now compare the result of the composition \((f \circ g)(x)\) with \(F(x)\). We find that \((f \circ g)(x) = (x+1)^5\) and \(F(x) = (x+1)^5\). Both expressions are exactly the same.

Key Concepts

Substitution

Substitution is an essential concept in mathematics, particularly in function composition. It involves replacing a variable in an expression with another expression or value. In function composition, this process allows us to evaluate functions by using the output of one function as the input for another.
Consider functions like those in our example:

• Function 1: \(g(x) = x + 1\)
• Function 2: \(f(x) = x^5\)

By substituting \(g(x)\) into \(f(x)\), you evaluate \(f(g(x))\). This means wherever there's an \(x\) in \(f(x)\), substitute with \(g(x) = x + 1\), resulting in the expression \((x + 1)^5\).
This method simplifies working with complex functions by breaking them down into manageable steps, building up from simpler expressions to more complicated ones. Keeping the process organized helps avoid mistakes and clarifies each step's logic.

Algebraic Functions

Algebraic functions are a type of function comprised of algebraic expressions calculated using basic arithmetic operations like addition, subtraction, multiplication, division, and raising to a power.
In our exercise, both \(f(x) = x^5\) and \(g(x) = x + 1\) are algebraic functions because they involve simple arithmetic processes applied to the variable \(x\).

• \(f(x)\) uses the power operation, derived only from multiplication and basic arithmetic.
• \(g(x)\) uses addition.

Algebraic functions are foundational in calculus and various fields of mathematics and science due to their straightforward operations which still offer wide applicability.
By understanding each function's algebraic structure, students can confidently engage with more complex problems involving combinations of these simpler expressions, honing their skills in both algebra and broader mathematical logic.

Comparing Expressions

When working with mathematical functions, comparing expressions involves determining whether two expressions are equivalent. It requires careful examination of the structure, operation, and results according to function rules.
In the original exercise, we compared two expressions:

• Result from substitution: \((f \circ g)(x) = (x + 1)^5\)
• Given function: \(F(x) = (x + 1)^5\)

By evaluating each step, we saw that both results simplify to the same expression. This confirms their equivalency, establishing that the statement is true.
This concept is crucial because it verifies that manipulated expressions using mathematical operations maintain their integrity and reflect the actual relationships between the elements involved. For students, practicing comparing expressions builds a strong foundation in problem-solving techniques and analytical thinking, empowering them to approach more complicated scenarios with confidence.