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Chapter 3: Problem 6

Classfy each function as odd, even, or neither. $$f(x)=x^{3}$$

Short Answer

Expert verified
The function \(f(x) = x^{3}\) is odd because \(f(-x) = -f(x)\).

Step by step solution

01

Identify the given function

The presented function is \(f(x)=x^{3}\).
02

Apply negative x to the function

Switch \(x\) in the function with \(-x\). This results in \(f(-x)=(-x)^{3}=-x^{3}\).
03

Compare with the original function

After substituting \(-x\) in the original function, it was found to be \(-x^{3}\), which is equivalent to \(-f(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even Functions
Understanding even functions is a fundamental concept in precalculus. An even function is one that satisfies the condition:
  • \( f(-x) = f(x) \) for all \( x \) in the domain of the function.
This means the graph of an even function is symmetric about the y-axis. A classic example of an even function is \( f(x) = x^2 \), since \( (-x)^2 = x^2 \). Visually, if you were to fold the graph over the y-axis, both sides would align perfectly.
It's important to test for evenness before analyzing any function further, as it can simplify calculations and help understand function behavior.
Function Classification
Function classification is all about determining whether a function is even, odd, or neither. This step is crucial as each function type has its distinct characteristics and symmetries.
  • Even functions: Have y-axis symmetry.
  • Odd functions: Have origin symmetry. They satisfy \( f(-x) = -f(x) \).
  • Neither: Functions that don't satisfy either condition.
Classifying functions helps streamline the analysis process, predict function behavior, and solve equations more efficiently.
When presented with a problem, start by testing for both even and odd characteristics to see where the function fits.
Precalculus Concepts
Precalculus is an essential stepping stone to more advanced mathematical concepts like calculus. It builds a strong foundation by covering a variety of fundamental ideas like functions, their properties, and graphing techniques.
  • Graphing: Identifying function types helps sketch accurate graphs.
  • Symmetry: Understanding symmetry in functions aids in further studies of geometric shapes and transformations.
  • Problem-solving: Learning these concepts eases the processor dealing with complex algebraic equations.
Mastering precalculus concepts prepares students for the challenges of calculus, where these foundational ideas are frequently revisited.
It's a phase that bridges the gap between algebra and calculus, making it indispensable for future mathematical success.

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Most popular questions from this chapter

Graph the function using a graphing utility, and find its zeros. $$g(x)=2 x^{5}+x^{4}-2 x-1$$

Solve the rational inequality. $$\frac{x+2}{x-1} \leq 0$$

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=x^{6}+4 x^{3}-3 x+7$$

Solve the rational inequality. $$\frac{-1}{3 x-1}>0$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{2}{x^{2}+4}$$
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