Evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3 c. g(-x)

Verified step by step guidance

1. Step 1: Understand the problem. You are given a function g(x) = x² + 2x + 3, and you are tasked with evaluating g(-x). This means substituting -x in place of x in the function.
2. Step 2: Substitute -x into the function. Replace every occurrence of x in g(x) with -x. The function becomes g(-x) = (-x)² + 2(-x) + 3.
3. Step 3: Simplify the first term (-x)². Recall that squaring a negative number results in a positive value, so (-x)² = x².
4. Step 4: Simplify the second term 2(-x). Multiply 2 by -x to get -2x.
5. Step 5: Combine all simplified terms. The function g(-x) simplifies to g(-x) = x² - 2x + 3.

Key Concepts

Function Evaluation

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

Simplification of Expressions

Simplification of expressions refers to the process of reducing a mathematical expression to its simplest form. This often involves combining like terms, factoring, or applying algebraic identities. For instance, after evaluating g(-x), one would simplify the resulting expression to make it easier to interpret or use in further calculations.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is crucial for evaluating and simplifying expressions involving them. In the given question, both functions are quadratic, which influences how we evaluate and simplify them.