Evaluate each function at the given values of the independent variable and simplify. g(x) = 3x^2 - 5x + 2 (a) g(0), (b) g(-2), (c) g(x-1), (d) g(-x)

Verified step by step guidance

1. Substitute the given value of x into the function g(x) = 3x^2 - 5x + 2 for each part of the problem.
2. For part (a), substitute x = 0 into the function: g(0) = 3(0)^2 - 5(0) + 2. Simplify the expression to find the result.
3. For part (b), substitute x = -2 into the function: g(-2) = 3(-2)^2 - 5(-2) + 2. Simplify the expression by squaring -2, multiplying, and combining like terms.
4. For part (c), substitute x = (x-1) into the function: g(x-1) = 3(x-1)^2 - 5(x-1) + 2. Expand (x-1)^2, distribute the constants, and combine like terms to simplify.
5. For part (d), substitute x = -x into the function: g(-x) = 3(-x)^2 - 5(-x) + 2. Simplify by squaring -x, distributing constants, and combining like terms.

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) = 3x^2 - 5x + 2, evaluating g(0) means replacing x with 0, resulting in g(0) = 3(0)^2 - 5(0) + 2 = 2.

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function g(x) = 3x^2 - 5x + 2 is a quadratic polynomial, characterized by its highest degree of 2, which influences its graph's shape and behavior.

Simplification

Simplification in mathematics refers to the process of reducing an expression to its simplest form. This can involve combining like terms, factoring, or performing arithmetic operations. In the context of evaluating g(x) for different inputs, simplification ensures that the final results are presented in the most concise and understandable manner.