In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3 b. g(x+2)

Verified step by step guidance

1. Start with the given function g(x) = x² + 2x + 3. To evaluate g(x+5), substitute x+5 for x in the function.
2. Replace every occurrence of x in the function with (x+5). This gives: g(x+5) = (x+5)² + 2(x+5) + 3.
3. Expand the squared term (x+5)² using the formula (a+b)² = a² + 2ab + b². This results in: g(x+5) = x² + 10x + 25 + 2(x+5) + 3.
4. Distribute the 2 across the (x+5) term: g(x+5) = x² + 10x + 25 + 2x + 10 + 3.
5. Combine all like terms (x² terms, x terms, and constant terms) to simplify the expression. This will give the final simplified form of g(x+5).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function to find the corresponding output. For example, in the function g(x) = x² - 10x - 3, evaluating g(2) means replacing x with 2, resulting in g(2) = 2² - 10(2) - 3. This process is fundamental for understanding how functions behave at different points.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The functions given, g(x) = x² - 10x - 3 and g(x) = x² + 2x + 3, are both quadratic polynomials, which are characterized by their highest degree of 2. Understanding the properties of polynomials, such as their shape and roots, is essential for evaluating and simplifying them.

Simplification of Expressions

Simplification of expressions involves reducing a mathematical expression to its simplest form, making it easier to understand and work with. This can include combining like terms, factoring, or expanding expressions. In the context of evaluating g(x+2) and g(x+5), simplification is crucial to express the results in a clear and concise manner, allowing for easier interpretation of the function's behavior.