In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x⁴ - x²+1 c. h (-x)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with evaluating the function h(x) = x⁴ - x² + 1 at h(-x). This means substituting -x into the function wherever x appears.
2. Step 2: Substitute -x into the function h(x). Replace every instance of x in the original function with -x. The function becomes h(-x) = (-x)⁴ - (-x)² + 1.
3. Step 3: Simplify each term. Recall that (-x)⁴ = x⁴ because raising a negative number to an even power results in a positive value. Similarly, (-x)² = x² for the same reason.
4. Step 4: Rewrite the simplified expression. After simplifying, the function becomes h(-x) = x⁴ - x² + 1.
5. Step 5: Compare h(-x) to h(x). Notice that h(-x) is identical to h(x), which means the function is even. This symmetry can be useful in understanding the behavior of the function.

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 every instance of x in the function h(x) 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, combined using addition, subtraction, and multiplication. The function h(x) = x^4 - x² + 1 is a polynomial of degree 4, which means its highest exponent is 4. 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 eliminating unnecessary components. After evaluating h(-x), the resulting expression may contain terms that can be combined or factored. Mastering simplification techniques is crucial for presenting clear and concise mathematical results.