Problem 3 - Algebra 2

Question

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ f(x)-g(x) $$

Step-by-Step Solution

Answer

So the result of subtracting \(g(x) = x^2\) from \(f(x) = 3x + 5\) is \(f(x) - g(x) = 3x + 5 - x^2\)

Step 1: Identify the functions

The two functions to be used in the operation are \(f(x) = 3x + 5\) and \(g(x) = x^2\).

Step 2: Perform the subtraction

Subtract the function \(g(x) = x^2\) from the function \(f(x) = 3x + 5\). This is performed as follows: \(f(x) - g(x) = (3x + 5) - x^2\).

Step 3: Simplify the result

In this particular case, no further simplification is needed, because there are no like terms to combine.

Key Concepts

Function Subtraction

Function subtraction is a method we use to subtract one function from another. It is an essential skill in algebra that helps us understand and manipulate functions. Given two functions, like \( f(x) \) and \( g(x) \), function subtraction is expressed as \( f(x) - g(x) \). Here's the simple process:

• First, identify both functions involved.
• Next, subtract the second function from the first.
• Finally, simplify the resulting expression.

This process helps in solving various mathematical problems and prepares you for more advanced topics in calculus and algebra.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. They are fundamental in algebra due to their simple to complex structures. A polynomial function can have:

• Constant terms like \(5\) or \(12\)
• Linear terms like \(3x\)
• Quadratic terms like \(x^2\)

The structure of a polynomial function makes it easy to perform arithmetic operations, such as addition or subtraction, because they follow specific rules based on the exponents of the variable. In our example, \( f(x) = 3x + 5 \) and \( g(x) = x^2 \), both are polynomial functions of different degrees, with linear and quadratic terms respectively. Understanding how to handle these terms is key in manipulating them correctly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators like addition or subtraction. They form the backbone of algebra, allowing us to express mathematical ideas in a concise form.An algebraic expression can include:

• Constants, which are fixed numbers (e.g., \(5\))
• Variables, which are symbols that represent numbers (e.g., \(x\))
• Operators, which indicate operations to be performed (e.g., +, -)

In function subtraction, we treat each function as an algebraic expression. When we subtract \( g(x) \) from \( f(x) \), we simply subtract the corresponding parts of each expression. For instance, here we're combining \( (3x + 5) \) and \( -x^2 \) to form a new algebraic expression: \( 3x + 5 - x^2 \). This simplified form allows us to further analyze or use the expression in future problems.