Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2 +1 b. h (-1)
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 31c
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 guidance1
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.
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.
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.
Step 4: Rewrite the simplified expression. After simplifying, the function becomes h(-x) = x⁴ - x² + 1.
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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Related Practice
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Textbook Question
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Textbook Question
Evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2+1 a. h (2)
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Textbook Question
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x4 - x2 +1 d. h (3a)
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