/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Determine whether \(f\) is even,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 9

Determine whether \(f\) is even, odd, or neither even nor odd. $$f(x)=\sqrt{x^{2}+4}$$

Short Answer

Expert verified
The function is even.

Step by step solution

01

Recall the Definitions

A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \). It is **odd** if \( f(-x) = -f(x) \) for all \( x \). If neither condition is satisfied, the function is neither odd nor even.
02

Calculate \( f(-x) \)

The given function is \( f(x) = \sqrt{x^2 + 4} \). Now, let's find \( f(-x) \): \[ f(-x) = \sqrt{(-x)^2 + 4} = \sqrt{x^2 + 4} \].
03

Compare \( f(x) \) and \( f(-x) \)

Since \( f(-x) = \sqrt{x^2 + 4} \) and \( f(x) = \sqrt{x^2 + 4} \), we have \( f(-x) = f(x) \). Thus, the function \( f \) is even.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Function Parity
Function parity is all about understanding how a function behaves when you input negative values. A function can be classified as even, odd, or neither based on this behavior. In simpler terms, this means:
  • Even Functions: If you plug in \(-x\) in place of \(x\) and the function remains unchanged, that is \(f(-x) = f(x)\), the function is called even.
  • Odd Functions: If plugging in \(-x\) results in the function being the negative of the original function, hence \(f(-x) = -f(x)\), the function is odd.
  • Neither: If neither condition is met, then the function is neither even nor odd.
Checking the parity of a function involves calculating \(f(-x)\) and comparing it to \(f(x)\). If they are identical, the function is even. If \(f(-x)\) is the negative of \(f(x)\), the function is odd. If neither, then it falls into the neither category. Parity is particularly useful for understanding the symmetry and behavior of functions on a graph.
Exploring Square Root Functions
Square root functions are a type of function that often involve \(\sqrt{x}\), where the output is the positive square root of the input value. In this context, we have the function \(f(x) = \sqrt{x^2 + 4}\). Let's break down what this means.
  • For any input \(x\), the function evaluates to the square root of \(x^2 + 4\).
  • Since squaring any real number (positive or negative) results in a positive outcome, \(x^2\) ensures that you're adding to a positive number 4 within the square root.
A key property of functions involving a square root is that the expression inside must be non-negative since square roots of negative numbers are not real numbers. This characteristic subtly impacts how the function behaves in terms of symmetry.
Understanding Symmetry in Functions
Symmetry in functions provides insight into how they map their inputs to their outputs visually on a graph. Functions with certain symmetries often make them easier to understand and analyze.One type of symmetry related to even functions is y-axis symmetry. This occurs when \(f(-x) = f(x)\), as with the example function \(f(x) = \sqrt{x^2 + 4}\). The fact that \(f(-x)\) results in exactly the same function means that the graph is the same on both sides of the y-axis.
  • For even functions, their graphs are mirror images across the y-axis.
  • Odd functions would show origin symmetry, meaning their graphs are mirrored when rotated 180 degrees around the origin.
By understanding these symmetries, you can predict how a function will behave and leverage these properties for further mathematical analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Precipitation in Seattle The average monthly precipitation (in inches) for Seattle is listed in the following table. (Note: April average is not given.) (a) Plot the average monthly precipitation. (b) Model the data with a quadratic function of the form \(f(x)=a(x-h)^{2}+k .\) Graph \(f\) and the data on the same coordinate axes. (c) Use \(f\) to predict the average rainfall in April. Compare your prediction with the actual value of 2.55 inches. $$\begin{array}{|l|c|} \hline \text { Month } & \text { Precipitation } \\ \hline \text { Jan. } & 5.79 \\ \hline \text { Feb. } & 4.02 \\ \hline \text { Mar. } & 3.71 \\ \hline \text { April } & \\ \hline \text { May } & 1.70 \\ \hline \text { June } & 1.46 \\ \hline \text { July } & 0.77 \\ \hline \text { Aug. } & 1.10 \\ \hline \text { Sept. } & 1.72 \\ \hline \text { Oct. } & 3.50 \\ \hline \text { Nov. } & 5.97 \\ \hline \text { Dec. } & 5.81 \\ \hline \end{array}$$

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=x^{2}-5 ; \quad g(x)=\frac{1}{4} x^{2}-5$$

(a) Use the quadratic formula to find the zeros of \(f .\) (b) Find the maximum or minimum value of \(f(x)\) (c) Sketch the graph of \(f\) $$f(x)=9 x^{2}+24 x+16$$

Find a composite function form for \(y\) $$y=\frac{1}{(x-3)^{6}}$$

Graph \(y=x^{3}-x^{1 / 3}\) and \(f\) on the same \(c o-\) ordinate plane, and estimate the points of intersection. $$f(x)=x^{2}-x-\frac{1}{4}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.