Problem 8 - Algebra Form and Function

Question

Use substitution to compose the two functions. $$ y=2 u^{2}+5 u+7 \text { and } u=3 x^{3} $$

Step-by-Step Solution

Answer

Answer: The composed function is \(y = 18x^6 + 15x^3 + 7\).

Step 1: Write down the given functions

The functions are given by: $$ y = 2u^2 + 5u + 7 \\ u = 3x^3 \\ $$

Step 2: Substitute the expression for u in terms of x

Replace \(u\) in the first function with the expression for \(u\) in terms of \(x\) given by the second function: $$ y = 2(3x^3)^2 + 5(3x^3) + 7 \\ $$

Step 3: Simplify the new function

Now simplify the function by performing the operations: $$ y = 2(9x^6) + 15x^3 + 7 \\ y = 18x^6 + 15x^3 + 7 \\ $$

Step 4: Write down the final composed function

The composed function in terms of \(x\) is: $$ y = 18x^6 + 15x^3 + 7 $$

Key Concepts

Substitution Method

The substitution method is a powerful algebraic technique commonly used in mathematics to solve equations and compose functions. When dealing with multiple functions, substitution simplifies the process by allowing us to replace one variable with another expression.

In the given exercise, substitution is used to compose two functions. You have a function for \( y \) that depends on \( u \), and another function that gives \( u \) in terms of \( x \).

To use substitution here, we take the expression of \( u \) from the second function and plug it into the first function wherever \( u \) appears. This way, we transform the function that originally depends on \( u \) into one that directly depends on \( x \).

The substitution method streamlines solving complex problems by eliminating intermediate variables, making computations simpler and more direct. This technique is widely used in calculus, algebra, and differential equations for function composition.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables.

In the exercise, the polynomial function is written in terms of the variable \( u \):

• \( y = 2u^2 + 5u + 7 \)

This polynomial is quadratic, with the highest power being 2. Each term in the polynomial can be separately calculated and combined to find the overall value.

After substitution, the function is converted, and we see the composition leading to another polynomial:

• \( y = 18x^6 + 15x^3 + 7 \)

This result is a polynomial in \( x \). The highest degree in this polynomial is 6, indicating a sextic polynomial. Polynomial functions are vital in mathematics due to their versatility and ease of manipulation, which helps in modeling real-world scenarios.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using a variety of algebraic techniques. These manipulations allow easier and more efficient computation of mathematical problems.

In the provided solution, algebraic manipulation is extensively used. First, we substitute \( u = 3x^3 \) into the polynomial \( y = 2u^2 + 5u + 7 \). Next, each term is multiplied and simplified:

• Compute \( (3x^3)^2 = 9x^6 \)
• Multiply by the coefficient to get \( 18x^6 \)
• Simplify \( 5(3x^3) \) to obtain \( 15x^3 \)

Finally, these terms are combined to form a new polynomial. Algebraic manipulation like this is fundamental in mathematics, as it helps to rewrite expressions in more usable forms, essential for solving equations and understanding function behaviors.