Evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 c. h (-x)

Verified step by step guidance

1. Step 1: Understand the problem. You are given a function h(x) = x³ − x + 1, and you are tasked with evaluating h(-x). This means substituting -x into the function wherever x appears.
2. Step 2: Substitute -x into the function. Replace every instance of x in h(x) with -x. The function becomes h(-x) = (-x)³ − (-x) + 1.
3. Step 3: Simplify the first term (-x)³. Recall that raising a negative number to an odd power results in a negative value. Therefore, (-x)³ simplifies to -x³.
4. Step 4: Simplify the second term -(-x). The double negative cancels out, so -(-x) simplifies to +x.
5. Step 5: Combine all the simplified terms. The function h(-x) now becomes h(-x) = -x³ + x + 1. This is the simplified form of the function when evaluated at -x.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate h(-x), we replace x in the function h(x) = x³ − x + 1 with -x. This process allows us to determine the output of the function for that particular input.

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers and combined using addition, subtraction, and multiplication. The function h(x) = x³ − x + 1 is a polynomial of degree three, which means its highest exponent is three. Understanding the structure of polynomial functions is essential for evaluating and simplifying them.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form by combining like terms and performing arithmetic operations. After evaluating h(-x), the next step is to simplify the resulting expression, which may involve rearranging terms or factoring. This process is crucial for presenting the final answer clearly and concisely.