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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1:36 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+1b. h (-1)

Verified step by step guidance

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

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

Calculate \((-1)^4\): Since any negative number raised to an even power is positive, \((-1)^4 = 1\).

Calculate \((-1)^2\): Similarly, \((-1)^2 = 1\).

Substitute these values back into the expression: \( h(-1) = 1 - 1 + 1 \).

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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(-1), we replace x in the function h(x) = x^4 - x² + 1 with -1. This process allows us to find 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 the highest power of x is 4. Understanding the structure of polynomial functions is essential for evaluating and simplifying them.

Simplification of Expressions

Simplification of expressions involves reducing a mathematical expression to its simplest form. After evaluating h(-1), we will combine like terms and perform any necessary arithmetic operations to simplify the result. This step is crucial for presenting the final answer clearly and concisely.