Which Rule Should Be Applied To Reflect $f(x) = X^3$ Over The $y$-axis?A. Multiply $f(y$\] By -1. B. Substitute $-x$ For $x$ And Simplify $f(-x$\]. C. Multiply $f(x$\] By -1. D. Switch The

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Introduction

When dealing with functions, reflecting them over the $y$-axis is a common operation in mathematics. This process involves changing the sign of the input variable, which can be achieved through various methods. In this article, we will explore the rules for reflecting a function over the $y$-axis and determine which one should be applied to the function $f(x) = x^3$.

Understanding Reflection Over the $y$-axis

To reflect a function over the $y$-axis, we need to change the sign of the input variable. This can be achieved by substituting $-x$ for $x$ in the function. However, there are different ways to approach this, and it's essential to understand the correct method to apply.

Option A: Multiply $f(y)$ by -1

One approach to reflecting a function over the $y$-axis is to multiply the function by $-1$. This method involves changing the sign of the entire function, but it's not the correct way to reflect a function over the $y$-axis. When we multiply a function by $-1$, we are essentially changing the sign of the output, not the input.

Option B: Substitute $-x$ for $x$ and Simplify $f(-x)$

The correct method to reflect a function over the $y$-axis is to substitute $-x$ for $x$ in the function and simplify. This approach involves changing the sign of the input variable, which is the correct way to reflect a function over the $y$-axis.

Option C: Multiply $f(x)$ by -1

As mentioned earlier, multiplying a function by $-1$ is not the correct way to reflect a function over the $y$-axis. This method changes the sign of the output, not the input, and is not applicable in this scenario.

Option D: Switch the $x$ and $y$ Variables

Switching the $x$ and $y$ variables is not a valid method for reflecting a function over the $y$-axis. This approach would result in a different function, not a reflection of the original function.

Applying the Correct Rule to $f(x) = x^3$

To reflect the function $f(x) = x^3$ over the $y$-axis, we need to substitute $-x$ for $x$ and simplify. This involves changing the sign of the input variable, which is the correct way to reflect a function over the $y$-axis.

Step-by-Step Solution

To apply the correct rule, follow these steps:

1. Substitute $-x$ for $x$ in the function $f(x) = x^3$.
2. Simplify the resulting expression.

Substituting $-x$ for $x$

When we substitute $-x$ for $x$ in the function $f(x) = x^3$, we get:

$f(-x) = (-x)^3$

Simplifying the Expression

To simplify the expression, we can use the property of exponents that states $(-a)^n = a^n$ when $n$ is an odd integer. In this case, $n = 3$, which is an odd integer.

$f(-x) = (-x)^3 = -x^3$

Conclusion

In conclusion, the correct rule to reflect $f(x) = x^3$ over the $y$-axis is to substitute $-x$ for $x$ and simplify. This involves changing the sign of the input variable, which is the correct way to reflect a function over the $y$-axis. The other options, multiplying the function by $-1$ or switching the $x$ and $y$ variables, are not valid methods for reflecting a function over the $y$-axis.

Final Answer

The final answer is: B. Substitute $-x$ for $x$ and simplify $f(-x)$.

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Introduction

In our previous article, we discussed the rules for reflecting functions over the $y$-axis. We explored the different methods for achieving this and determined that substituting $-x$ for $x$ and simplifying is the correct approach. In this article, we will answer some frequently asked questions related to reflecting functions over the $y$-axis.

Q: What is the purpose of reflecting a function over the $y$-axis?

A: Reflecting a function over the $y$-axis is a common operation in mathematics that involves changing the sign of the input variable. This can be useful in various applications, such as graphing functions, solving equations, and analyzing functions.

Q: How do I reflect a function over the $y$-axis?

A: To reflect a function over the $y$-axis, you need to substitute $-x$ for $x$ in the function and simplify. This involves changing the sign of the input variable, which is the correct way to reflect a function over the $y$-axis.

Q: What is the difference between reflecting a function over the $y$-axis and multiplying it by $-1$?

A: Reflecting a function over the $y$-axis involves changing the sign of the input variable, while multiplying a function by $-1$ changes the sign of the output. These are two different operations, and multiplying a function by $-1$ is not the correct way to reflect a function over the $y$-axis.

Q: Can I switch the $x$ and $y$ variables to reflect a function over the $y$-axis?

A: No, switching the $x$ and $y$ variables is not a valid method for reflecting a function over the $y$-axis. This approach would result in a different function, not a reflection of the original function.

Q: How do I apply the correct rule to a function?

A: To apply the correct rule, follow these steps:

1. Substitute $-x$ for $x$ in the function.
2. Simplify the resulting expression.

Q: What if the function has multiple variables?

A: If the function has multiple variables, you need to substitute $-x$ for $x$ in each variable and simplify. For example, if the function is $f(x, y) = x^2y$, you would substitute $-x$ for $x$ and $-y$ for $y$ to get $f(-x, -y) = (-x)^2(-y) = x^2y$.

Q: Can I reflect a function over the $y$-axis if it has a fractional exponent?

A: Yes, you can reflect a function over the $y$-axis even if it has a fractional exponent. The correct rule still applies, and you need to substitute $-x$ for $x$ and simplify.

Q: How do I reflect a function over the $y$-axis if it has a trigonometric function?

A: To reflect a function over the $y$-axis that has a trigonometric function, you need to substitute $-x$ for $x$ and simplify. For example, if the function is $f(x) = \sin x$, you would substitute $-x$ for $x$ to get $f(-x) = \sin (-x) = -\sin x$.

Conclusion

In conclusion, reflecting functions over the $y$-axis is a common operation in mathematics that involves changing the sign of the input variable. By following the correct rule and substituting $-x$ for $x$ and simplifying, you can reflect a function over the $y$-axis. We hope this Q&A article has provided you with a better understanding of this concept and how to apply it in different situations.

Final Answer

The final answer is: Substitute $-x$ for $x$ and simplify.