Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

2. Graphs of Equations

Two-Variable Equations

College Algebra

2. Graphs of Equations

Two-Variable Equations

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1:51 minutes

Problem 31

Textbook Question

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x^4 - x²+1c. h (-x)

Verified step by step guidance

Identify the given function: \( h(x) = x^4 - x^2 + 1 \).

To evaluate \( h(-x) \), substitute \(-x\) for \(x\) in the function.

This gives: \( h(-x) = (-x)^4 - (-x)^2 + 1 \).

Simplify each term: \((-x)^4 = x^4\) and \((-x)^2 = x^2\).

Combine the simplified terms: \( h(-x) = x^4 - x^2 + 1 \).

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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 analyze how the function behaves with different inputs, which is essential for understanding its properties.

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 indicates its highest exponent. Understanding polynomial functions is crucial for evaluating and simplifying expressions, as they exhibit specific behaviors such as continuity and differentiability.

Simplification of Expressions

Simplification of expressions involves reducing a mathematical expression to its simplest form. This can include combining like terms, factoring, or applying algebraic identities. After evaluating h(-x), simplifying the resulting expression helps in clearly understanding the function's characteristics and making further calculations or comparisons easier.