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Chapter 3: Problem 60

Give an example of a function \(g\) with the property that \(g(x)=g(-x)\) for every real number \(x\)

Short Answer

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Question: Provide an example of a function g that satisfies the property g(x) = g(-x) for every real number x. Answer: An example of a function g with the property g(x) = g(-x) for every real number x is g(x) = 3x^2.

Step by step solution

01

Understand the Given Information

Notice that the given property g(x) = g(-x) indicates that g is an even function. An even function is symmetric about the y-axis, meaning that its graph is unchanged if it is reflected across this axis.
02

Example of an Even Function

A simple example of an even function is a quadratic function of the form g(x) = ax^2, where a is a non-zero real constant. When x is replaced by -x, g(-x) = a(-x)^2 = ax^2 = g(x). Therefore, any quadratic function of this form is an example of an even function satisfying the given property.
03

Provide the Even Function Example

An example of a function g with the property g(x) = g(-x) for every real number x is g(x) = 3x^2, which is a quadratic function of the form g(x) = ax^2 and satisfies the given property.

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Key Concepts

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

Function Symmetry
Function symmetry is a crucial concept in mathematics, especially when analyzing even and odd functions. When a function is symmetric, it has a balanced appearance across a particular line or point.

For even functions, like the one described in the original exercise, symmetry occurs over the y-axis. This means if you pick any point on the graph and reflect it across the y-axis, it will land on another point of the graph.

This symmetry property is mathematically expressed as:
  • For every function value at a point \(x\), the function value at \(-x\) is the same, i.e., \(f(x) = f(-x)\).
Understanding this concept helps us visually identify and analyze the behavior of functions quickly.
Quadratic Functions
Quadratic functions are polynomial functions of degree 2 and have the general form \(f(x) = ax^2 + bx + c\). They are among the simplest even functions when \(b = 0\), which ensures they are symmetric around the y-axis.

In our example, a specific form \(g(x) = ax^2\) shows an even quadratic function. Here:
  • \(a\) is a constant that controls the width and direction of the parabola.
  • The lack of a linear term \(bx\) assures perfect y-axis symmetry.
When \(g(x) = 3x^2\), we have a parabola opening upwards, and this function is an excellent demonstration of a quadratic that meets the symmetry condition \(g(x) = g(-x)\).
Graph Properties
Graph properties of quadratic functions reveal many insights into their structure and behavior. These functions typically graph as parabolas, which have distinct characteristics:

  • Vertex: This is the highest or lowest point on the parabola, representing the function's maximum or minimum value.
  • Axis of symmetry: For our specific quadratic of the form \(ax^2\), this line of symmetry is the y-axis itself.
  • Direction of opening: Controlled by the coefficient \(a\), the parabola can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
The function \(g(x) = 3x^2\) has these graph properties, making it very predictable and easy to work with in calculations and graphs.
Y-Axis Symmetry
Y-axis symmetry is a defining feature of even functions. When a function, like \(g(x) = 3x^2\), displays this symmetry, the graph appears identical on both sides of the y-axis.

This symmetry means:
  • Reflecting the graph over the y-axis will not change its appearance.
  • For every point \((x, g(x))\), there is a corresponding point \((-x, g(x))\).
This conceptual understanding of y-axis symmetry makes it easier to graph functions and notice patterns in their behavior. It allows for predicting the function's values just by knowing half of the graph, a useful property for both plotting by hand and analyzing data.

Function symmetry, especially y-axis symmetry, simplifies many mathematical problems and helps in the study of function behaviors and transformations.

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Most popular questions from this chapter

Write the given function as the composite of two functions, neither of which is the identity function, as in Examples 6 and 7 . (There may be more than one way to do this.) $$h(x)=\left(7 x^{3}-10 x+17\right)^{7}$$

Give an example of two different functions \(f\) and \(g\) that have all of the following properties: $$ f(-1)=1=g(-1) \quad \text { and } \quad f(0)=0=g(0) $$ and \(\quad f(1)=1=g(1)\)

If \(f(x)=x^{3}+c x^{2}+4 x-1\) for some constant \(c\) and \(f(1)=2,\) find \(c .[\text {Hint: Use the rule of } f \text { to compute } f(1) .]\)

Sketch the graph of a function \(f\) that satisfies these five conditions: (i) \(f(-1)=2\) (ii) \(f(x) \geq 2\) when \(x\) is in the interval \(\left(-1, \frac{1}{2}\right)\) (iii) \(f(x)\) starts decreasing when \(x=1\) (iv) \(f(3)=3=f(0)\) (v) \(f(x)\) starts increasing when \(x=5\) [Note: The function whose graph you sketch need not be given by an algebraic formula.]

Use algebra to find the inverse of the given one-to-one function. $$f(x)=\left(x^{5}+1\right)^{3}$$
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