Problem 49 - Algebra 2

Question

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ f(x)-g(x)+10 $$

Step-by-Step Solution

Answer

The result of the operation \(f(x)-g(x)+10\) is \(-x^{2}+5x+13\).

Step 1: Identify the Functions

The functions to work with are \(f(x) = 2x+5\) and \(g(x) = x^{2}-3x+2\).

Step 2: Perform Subtraction f(x)-g(x)

To subtract one function from another means to subtract their respective expressions separately. Do \(f(x)-g(x)= (2x+5)-(x^{2}-3x+2)= -x^{2}+5x+3\).

Step 3: Add the Constant

Perform the operation \(f(x)-g(x)+10\) by adding 10 to the result from the previous step: \(-x^{2}+5x+3+10 = -x^{2}+5x+13\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of algebraic expression that include terms in the form of an independent variable, raised to a whole number power, and multiplied by a coefficient. These are fundamental concepts in algebra and calculus.
Polynomials can vary in degree, which is determined by the highest power of the variable in the expression. For example:

• In the function \(g(x) = x^2 - 3x + 2\), the degree is 2, since the highest power of \(x\) is 2.
• In \(f(x) = 2x + 5\), the degree is 1, as the highest power is 1.

Polynomial functions can take various forms:

• Linear functions (degree 1): \(f(x) = mx + b\)
• Quadratic functions (degree 2): \(g(x) = ax^2 + bx + c\)
• Cubic functions (degree 3): \(h(x) = ax^3 + bx^2 + cx + d\)

Understanding how to recognize and manipulate these functions is crucial for many areas of mathematics, including equation solving and calculus.

Function Subtraction

Function subtraction is a basic operation where you subtract one function's output from another's.
Given two functions, such as \(f(x)\) and \(g(x)\), subtraction can be expressed as:

• \((f - g)(x) = f(x) - g(x)\)

In practice, you substitute in the expressions for \(f(x)\) and \(g(x)\), and perform the subtraction on their algebraic expressions.
For example, subtracting \(g(x) = x^2 - 3x + 2\) from \(f(x) = 2x + 5\) gives:

• \((2x + 5) - (x^2 - 3x + 2)\)
• This simplifies to \(-x^2 + 5x + 3\)

The subtraction of functions follows the rules of algebra, meaning that you distribute negative signs and combine like terms to simplify the final expression.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations (addition, subtraction, multiplication, division, and exponentiation) arranged together. These expressions form the foundation of algebra.
Expressions can vary widely in complexity and form. For the functions given in the exercise:

• \(f(x) = 2x + 5\) is a linear expression.
• \(g(x) = x^2 - 3x + 2\) is a quadratic expression.

Key aspects of working with algebraic expressions include:

• Combining like terms: terms with the same variable and exponent.
• Applying the distributive property: spreading operations over terms in parentheses.
• Rearranging terms to simplify expressions.

When performing operations like addition or subtraction on algebraic expressions, it's essential to pay attention to the signs and coefficients of each term. This ensures that the resulting expression is correctly simplified and reflects the intended mathematical operation.