Evaluate each function at the given values of the independent variable and simplify. h(x) = x⁴ - x²+1 a. h (2)

Verified step by step guidance

1. Step 1: Understand the problem. The function h(x) = x⁴ - x² + 1 is given, and we are tasked with evaluating it at x = 2. This means substituting x = 2 into the function and simplifying the resulting expression.
2. Step 2: Substitute x = 2 into the function. Replace every occurrence of x in h(x) with 2. The expression becomes h(2) = 2⁴ - 2² + 1.
3. Step 3: Simplify the powers of 2. Calculate 2⁴ (which is 2 raised to the power of 4) and 2² (which is 2 raised to the power of 2).
4. Step 4: Perform the subtraction and addition. Subtract the value of 2² from the value of 2⁴, then add 1 to the result.
5. Step 5: Write the simplified result. After completing the arithmetic operations, you will have the value of h(2).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate h(2), you replace x in the function h(x) = x^4 - x² + 1 with 2, allowing you to compute the output of the function at that point.

Polynomial Functions

The function h(x) = x^4 - x² + 1 is a polynomial function, which is a mathematical expression consisting of variables raised to whole number powers and coefficients. Understanding polynomial functions is crucial as they exhibit specific properties, such as continuity and differentiability, and can be analyzed using algebraic techniques.

Simplification

Simplification in mathematics refers to the process of reducing an expression to its simplest form. After evaluating the function h(2), you will need to combine like terms and perform arithmetic operations to arrive at a final, simplified result, which is essential for clear communication of the answer.