Problem 32 - Algebra 2

Question

Find \([g \circ h](x)\) and \([h \circ g](x)\) $$ \begin{array}{l}{g(x)=2 x} \\ {h(x)=x^{3}+x^{2}+x+1}\end{array} $$

Step-by-Step Solution

Answer

\([g \circ h](x) = 2x^3 + 2x^2 + 2x + 2\) and \([h \circ g](x) = 8x^3 + 4x^2 + 2x + 1\).

Step 1: Understanding the Compositions

We need to find the compositions \([g \circ h](x)\) and \([h \circ g](x)\). This requires plugging one function into the other. \([g \circ h](x)\) means plugging the output of \(h(x)\) into \(g(x)\), and \([h \circ g](x)\) means plugging the output of \(g(x)\) into \(h(x)\).

Step 2: Compute \([g \circ h](x)\)

Substitute \(h(x)\) into \(g(x)\):\[g(h(x)) = g(x^3 + x^2 + x + 1) = 2(x^3 + x^2 + x + 1)\].Distribute the 2:\[g(h(x)) = 2x^3 + 2x^2 + 2x + 2\].

Step 3: Compute \([h \circ g](x)\)

Substitute \(g(x)\) into \(h(x)\):\[h(g(x)) = h(2x) = (2x)^3 + (2x)^2 + (2x) + 1\].Simplify the expression:\[h(g(x)) = 8x^3 + 4x^2 + 2x + 1\].

Key Concepts

Algebra

Algebra forms the basis of solving mathematical equations by utilizing symbols and abstract concepts to represent numbers and values. Algebraic expressions use variables (like \(x\) or \(y\)) to stand in for unknown values. This allows us to create equations that can describe a wide range of scenarios. Here we appreciate not just numbers but the relationships between them, expressed through equations and expressions.
To solve these algebraic problems, understanding operations such as addition, subtraction, multiplication, and division is crucial. These operations help simplify complex expressions and solve equations step by step.
In function composition problems, algebra is super important because it helps us manipulate expressions as we compose one function with another. Knowing algebraic operations lets us rearrange and solve the resulting expressions effectively. By doing this, we're able to find the resulting outputs of function compositions, as shown in this exercise.

Polynomials

Polynomials are an integral part of algebra and are mathematical expressions involving sums of powers in one or more variables multiplied by coefficients. For example, in the given problem, \(h(x) = x^3 + x^2 + x + 1\) is a polynomial.
Polynomials can have various degrees based on the highest power of the variable. Here, the degree is 3, as the highest power of \(x\) is \(x^3\).
When working with polynomials, key arithmetic operations include:

• Addition and subtraction of like terms
• Multiplication to expand products
• Division, often to simplify expressions

In this function composition problem, understanding how to handle polynomial expressions is vital. Each function's output is a polynomial, and when composing the functions, we combine them by substituting one polynomial into another, requiring careful algebraic manipulation.

Mathematical Functions

Understanding mathematical functions is essential in many areas of math, including algebra and calculus. A function is a relation between a set of inputs and a set of possible outputs. Each input is related to exactly one output.
In this problem, we dealt with two functions:

• \(g(x) = 2x\)
• \(h(x) = x^3 + x^2 + x + 1\)

Function composition is a method used to combine two functions. For function composition, we "plug" one function into another. This essentially means taking the output of one function and using it as the input to another.
The notation

• \([g \circ h](x) = g(h(x))\)
• \([h \circ g](x) = h(g(x))\)

indicates function composition. By understanding how inputs and outputs interact, you can predict the behavior of complex functions created through composition, like those we solved for in this example.