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Chapter 1: Problem 2

$$ \mathrm{T} / \mathrm{F}: \text { If } g(x)=x^{2}, \text { then } g(2)=g(-2) $$

Short Answer

Expert verified
True: \( g(2) = g(-2) \) because both equal 4.

Step by step solution

01

Understand the Problem

We need to evaluate the expression \( g(x) = x^2 \) at \( x = 2 \) and \( x = -2 \), and determine if \( g(2) = g(-2) \).
02

Evaluate g(2)

Substitute \( x = 2 \) into the function: \( g(2) = (2)^2 = 4 \). So, \( g(2) = 4 \).
03

Evaluate g(-2)

Substitute \( x = -2 \) into the function: \( g(-2) = (-2)^2 = 4 \). Thus, \( g(-2) = 4 \).
04

Compare g(2) and g(-2)

We found that \( g(2) = 4 \) and \( g(-2) = 4 \). Since the values are equal, \( g(2) = g(-2) \).

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Key Concepts

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

Function Evaluation
In mathematics, evaluating a function is the process of substituting a given value into the function to determine its corresponding output. For the quadratic function given, \( g(x) = x^2 \), evaluating the function at specific points helps us understand how the function behaves with different inputs.

To evaluate \( g(x) \) at a point, follow these steps:
  • Substitute the specific value of \( x \) into the function's equation.
  • Perform any necessary arithmetic operations, such as squaring the number.
  • Record the outcome, which is the value of \( g(x) \) at that point.

For instance, substituting \( x = 2 \) into \( g(x) = x^2 \), results in \( g(2) = 2^2 = 4 \). Similarly, substituting \( x = -2 \) gives \( g(-2) = (-2)^2 = 4 \). Functions can thus reveal their properties and patterns through various evaluations.
Symmetry in Functions
Symmetry is a key concept in understanding the behavior of functions. A function is symmetric if its graph can be reflected identity-wise across a line or a point. For quadratic functions like \( g(x) = x^2 \), symmetry is observed distinctly.

Quadratic functions often have symmetry about the y-axis. If a function exhibits this symmetry, it means \( g(x) = g(-x) \) for all \( x \). Our current function, \( g(x) = x^2 \), demonstrates this perfectly:
  • \( g(2) = 4 \)
  • \( g(-2) = 4 \)

This tells us that the function's graph is identical to the left and right of the y-axis, confirming that \( g(x) = g(-x) \). This axis symmetry is a hallmark of even functions, to which quadratic functions belong.
Squares of Numbers
Understanding how to square a number is foundational in mathematics. The act of squaring a number means multiplying the number by itself. For instance, \( 4^2 = 4 \times 4 = 16 \) or \( (-3)^2 = (-3) \times (-3) = 9 \).

Here are a few key points about the squares of numbers:
  • The square of positive or negative numbers always results in a non-negative value.
  • Squaring is neither commutative nor associative, but it exhibits certain predictable patterns.
  • Every positive number has two square roots: one positive and one negative. For instance, \( \sqrt{16} = 4 \) and \( -4 \).

In our function \( g(x) = x^2 \), squaring both \( 2 \) and \( -2 \) yields \( 4 \), demonstrating that the outcome of squaring is consistently non-negative, which plays into the property of symmetry we observe.

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Most popular questions from this chapter

Is \(3 x(2 x+3)^{-5 / 3}\) in radical or exponential form?

Evaluate the described mathematical statement, or determine how the described changes affect other variables in the statement as appropriate. Consider a square-based box whose height is half the length of the sides for the base. If the surface area of the base is \(16 \mathrm{ft}^{2}\), what is the volume of the box?

Evaluate the given statement. $$ 4^{\log _{2}\left(2^{2}\right)} $$

Evaluate the described mathematical statement, or determine how the described changes affect other variables in the statement as appropriate. The velocity of periodic waves, \(v,\) is given by \(v=\lambda f\) where \(\lambda\) is the length of the waves and \(f\) is the frequency of the waves. If the wavelength is held constant while the frequency is tripled, what happens to the velocity of the waves? Be as descriptive as possible.

In exercises \(6-12,\) expand and simplify the given expressions. $$ \left(x^{2}+3 x-1\right)(4 x)+\left(2 x^{2}-5\right)(2 x+3) $$
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