Question

Use substitution to compose the two functions. $$ y=u^{2}+u+1 \text { and } u=x^{2} $$

Short answer

Question: Find the composite function of $y = u^2 + u + 1$ and $u = x^2$. Answer: The composite function is $y = x^4 + x^2 + 1$.

Step-by-step solution

Write down the given functions

We are given the two functions: $$ y = u^2 + u + 1 \text{ and } u = x^2. $$

Substitute the second function into the first function

To find the composite function, substitute the second function (\(u = x^2\)) into the first function: $$ y = (x^2)^2 + (x^2) + 1. $$

Simplify the composite function

Now, simplify the composite function by performing the operations on the terms and combining like terms if any: $$ y = x^4 + x^2 + 1. $$ Thus, the composite function is: $$ y = x^4 + x^2 + 1. $$

Key concepts

Substitution Method

The substitution method is a mathematical technique used to simplify equations and expressions by replacing a variable with another expression. Think of it as a way of switching one part of a problem with something else that has the same value but is expressed differently.
In the given exercise, we used the substitution method to find a composite function. Here's how:

• We had two functions, one in terms of \(u\) and the second defining \(u\) in terms of \(x\).
• By substituting \(u = x^2\) into the first function \(y = u^2 + u + 1\), we replace all instances of \(u\) with \(x^2\).

This method helps to merge multiple functions into one single expression.

Composite Function

A composite function, in simple words, refers to a function that is made up by combining two functions. The result is a new function where one function's output becomes the input of another.
In our scenario, we combined the functions \(y = u^2 + u + 1\) and \(u = x^2\). Therefore, by substituting \(x^2\) for \(u\), we created a new single function \(y = x^4 + x^2 + 1\).

• The inner function, \(u = x^2\), was inserted into the outer function, \(y = u^2 + u + 1\).
• Normally, the notation \((f \circ g)(x)\) is used to indicate a composite function where \(f(u)\) is the outer and \(g(x)\) is the inner.

Creating composite functions can help simplify complex relationships between different variables and reveal insights into how they interact.

Polynomial Simplification

Polynomial simplification is an essential process in algebra where an expression with several terms is written in its simplest form. This involves operations such as expanding brackets, combining like terms, and simplifying coefficients.
In our composite function, we ended with \(y = (x^2)^2 + x^2 + 1\). Simplification occurs as follows:

• The term \((x^2)^2\) was expanded to \(x^4\) through multiplication of powers since \((a^m)^n = a^{m \cdot n}\).
• Finally, no like terms were present, so the terms were combined as they were: \(x^4 + x^2 + 1\).

Simplifying polynomials makes them easier to work with and understand, especially when solving equations or graphing the function.