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Chapter 3: Problem 3

Let \(g(x)=x^{3}\) and find \(g(1), g(-1), g(2)\) \(g(-2), g(3)\) and \(g(-3)\). What do you notice about your results?

Short Answer

Expert verified
Values of \(g(x)\) for opposites \(x\) are symmetric with opposite signs.

Step by step solution

01

- Define the function

The given function is defined as follows: \[ g(x) = x^3 \]. This means for any input value of \(x\), we need to cube this value to find \(g(x)\).
02

- Find \(g(1)\)

Substitute \(x = 1\) into the function: \[ g(1) = 1^3 = 1 \]
03

- Find \(g(-1)\)

Substitute \(x = -1\) into the function: \[ g(-1) = (-1)^3 = -1 \]
04

- Find \(g(2)\)

Substitute \(x = 2\) into the function: \[ g(2) = 2^3 = 8 \]
05

- Find \(g(-2)\)

Substitute \(x = -2\) into the function: \[ g(-2) = (-2)^3 = -8 \]
06

- Find \(g(3)\)

Substitute \(x = 3\) into the function: \[ g(3) = 3^3 = 27 \]
07

- Find \(g(-3)\)

Substitute \(x = -3\) into the function: \[ g(-3) = (-3)^3 = -27 \]
08

- Analyze results

Noticing the results for positive and negative values of \(x\), the function \(g(x) = x^3\) produces symmetric values about zero but with opposite signs. Specifically, if \(g(x)\) is \(a\), then \(g(-x)\) is \(-a\).

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Key Concepts

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

Cubed Functions
A cubed function is a type of polynomial function where the highest power of the variable is 3. Mathematically, it is represented as \(f(x) = x^3\). This means that for any input value of \(x\), you multiply \(x\) by itself three times: \(x \times x \times x = x^3\).

For example, if you input \(x = 2\), the output will be \(2^3 = 8\). Similarly, if the input is \(-2\), the output will be \((-2)^3 = -8\).

Cubing both positive and negative numbers results in values that are symmetrical about zero but have opposite signs. This is because a negative number multiplied three times retains its negative sign, while a positive number retains its positive sign.
Function Evaluation
Function evaluation is the process of finding the output of a function for a particular input value. To evaluate a function, you substitute the given input into the function's formula and simplify.

For the function \(g(x) = x^3\), to find \(g(1)\), you substitute \(1\) in place of \(x\):
\[ g(1) = 1^3 = 1 \]
Similarly, to find \(g(-1)\), \(-1\) is substituted for \(x\):
\[ g(-1) = (-1)^3 = -1 \]
So, evaluating functions is all about substituting the input value into the formula and solving it step by step.
Symmetry in Functions
Symmetry in functions refers to the property of a function's graph where one half is a mirror image of the other half. For the function \(g(x) = x^3\), this type of symmetry is known as odd symmetry.

In this exercise, evaluating \(g(x)\) for positive and negative values of \(x\) showed that \(g(-x) = -g(x)\). This means that the function is symmetric about the origin, since for any point \((x, y)\) on the graph, there is a corresponding point \((-x, -y)\).

For instance, when \(x = 2\), \(g(2) = 8\) and when \(x = -2\), \(g(-2) = -8\). This opposite-sign property holds for all values of \(x\), confirming the odd symmetry of \(g(x) = x^3\).

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Most popular questions from this chapter

By using your calculator, estimate the following limits: a \(\lim _{x \rightarrow 0} 5^{x}\) b \(\lim _{x \rightarrow \infty} 2.71828^{-x}\)

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The function \(f\) is defined by \(f(x)=\frac{6}{3-x} \quad(x \neq 3)\) Find i \(f^{-1}\) ii \(f \circ f^{-1}\) iii \(f^{-1} \circ f\)
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