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

Verified step by step guidance

1. Substitute \( x + 2 \) into the function \( g(x) = x^2 - 10x - 3 \) to get \( g(x+2) = (x+2)^2 - 10(x+2) - 3 \).
2. Expand \( (x+2)^2 \) to get \( x^2 + 4x + 4 \).
3. Distribute \( -10 \) across \( (x+2) \) to get \( -10x - 20 \).
4. Combine all the terms: \( x^2 + 4x + 4 - 10x - 20 - 3 \).
5. Simplify the expression by combining like terms.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, we need to replace 'x' in the function g(x) = x² - 10x - 3 with (x + 2). This process allows us to determine the output of the function for that particular input.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. After substituting (x + 2) into the function, we will expand the resulting expression and combine like terms to simplify it. This step is crucial for making the expression easier to work with and understand.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The function g(x) = x² - 10x - 3 is a quadratic function, and understanding its properties, such as its graph and vertex, can provide insights into its behavior and the significance of its outputs.