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

Guida verificata passo dopo passo

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.

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.

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.

Step 4: Simplify the second term -(-x). The double negative cancels out, so -(-x) simplifies to +x.

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.

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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.

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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.

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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.

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