Textbook Question
Graph each piecewise-defined function.
6
views
2. Graphs of Equations
Two-Variable Equations
1:51 minutesProblem 31c
Textbook Question
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey 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.
Recommended video:
4:26
Evaluating Composed Functions
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.
Recommended video:
06:04Introduction to Polynomial Functions
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.
Recommended video:
Guided course
05:09Introduction to Algebraic Expressions
Related Videos
Related Practice