Problem 74 - Algebra 2

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ (f-g)(x) $$

Step-by-Step Solution

Answer

The simplified form of \((f-g)(x)\) is \(-x^2 + 3x - 3\).

Step 1: Identify the functions

The functions given are \(f(x)=3x-2\) and \(g(x)=x^2+1\).

Step 2: Carry out the operation

The operation \(f-g\) is to be performed. This means we will subtract function \(g(x)\) from function \(f(x)\). So, \((f-g)(x) = f(x) - g(x)\) which results into \((f-g)(x) = (3x-2) - (x^2+1)\).

Step 3: Simplify the expression

Simplify the expression \((f-g)(x) = (3x-2) - (x^2+1)\). This can be done by distributing the negative sign to each term inside the parenthesis resulting in \((f-g)(x) = 3x - 2 - x^2 -1.\)

Step 4: Further Simplify the Expression

To further simplify, we group like terms together. Resulting into \((f-g)(x) = -x^2 + 3x - 3\). Here, the properties of real numbers were used to move the terms around freely. The properties used include the commutative and associative properties of addition and subtraction.

Key Concepts

Subtraction of Functions

Subtracting functions might sound complicated at first, but it's quite straightforward once you break it down. When we talk about subtracting one function from another, we simply mean finding the difference between their outputs for any given input. This is denoted as

• \((f-g)(x) = f(x) - g(x)\)

Imagine you have two functions, where one gives you a certain number of cookies, and the other one takes some away. The operation of subtraction reveals how many cookies you're left with. For the functions provided,

• \(f(x)=3x-2\)
• \(g(x)=x^2+1\)

Subtraction involves calculating each term separately, ensuring that you maintain the order of operations.

• First, rewrite the expression, which gives us \((f-g)(x) = (3x-2) - (x^2+1)\).

Remember, subtraction means taking away every part of the second function, and crucially, it involves changing the signs of the elements being subtracted due to the distributive property of mathematics.

Properties of Real Numbers

The properties of real numbers are essential rules that allow us to manipulate and simplify expressions smoothly. These properties include:

• Commutative Property: This is about the order of addition and multiplication. For instance, \(a + b = b + a\).
• Associative Property: This covers grouping changes in addition or multiplication. So, \((a + b) + c = a + (b + c)\).
• Distributive Property: This property helps expand expressions, such as \(a(b + c) = ab + ac\).

When subtracting functions, we extensively use the commutative and associative properties. In our work to simplify

• \((3x-2) - (x^2 + 1)\),

we rearrange terms and change their signs, knowing these properties maintain equality. These properties allow us to "move terms around" without changing the result, as long as we're careful with signs, especially during subtraction when distributing negative signs across parentheses.

Polynomial Functions

Polynomial functions are a primary type of mathematical expression that include variables raised to whole number exponents. They are composed of many ('poly') terms, each being a constant multiplied by a variable raised to an exponent, like

• \(ax^n + bx^{n-1} + cx^{n-2} + ... + k\).

In this exercise, both given functions are parts of polynomials:

• \(f(x) = 3x - 2\)
• \(g(x) = x^2 + 1\)
• The resulting function \((-x^2 + 3x - 3)\)

is also a polynomial. A polynomial function's degree is determined by the highest power of the variable present. Here,

• \(g(x) = x^2 + 1\) is of degree 2 because the highest power of \(x\) is 2.

This expression,

• \(-x^2 + 3x - 3\),

simplifies to show how polynomial functions elegantly handle subtraction through combining like terms and respecting their arithmetic properties.