Problem 17 Identify the equations that defi... [FREE SOLUTION]

Chapter 1: Problem 17

Identify the equations that define $y$ as a function of $x .$ $$y=\sqrt[3]{x}$$

Short Answer

Expert verified

The function that describes $y$ as a function of $x$ is $y = \sqrt[3]{x}$. It can also be written in function notation as $f(x) = \sqrt[3]{x}$.

Step by step solution

Identifying the Function

The given expression is: $ y = \sqrt[3]{x} $. This is a function because for every input $x$, there is one and only one output $y$. The function describes $y$ as the cube root of $x$. This means, for a given $x$, $y$ is the number that, when cubed, equals $x$.

Function Notation

This function could be written in function notation as $f(x) = \sqrt[3]{x}$. This emphasizes that $y$ is a function of $x$, meaning its value depends on the value of $x$.

Example of Function Evaluation

For example, if $x= 8$, then $y = f(x) = \sqrt[3]{8} = 2$. This demonstrates how $y$ depends on $x$ in this function.

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Key Concepts

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

Function Notation

Function notation is a way to name a function in a concise manner. Instead of writing equations like $y = \sqrt[3]{x}$, we use a notation that links the function directly to its variable. In this case, we can write $f(x) = \sqrt[3]{x}$.
This means "$f$ of $x$" where $f$ is the function, and $x$ is the input.
Function notation emphasizes that the output, often represented by $y$, depends on the value of $x$.
Here are some key points about function notation:

$f(x)$ is read as 鈥榎(f\) of $x$鈥�, meaning the function $f$ applied to $x$.
It provides clarity that $y$ is defined by a specific rule or equation involving $x$.
Many functions can use the same letter, like $f$, to identify them, but they are defined by the specific expression following the equals sign.

This method is particularly useful in math because it allows functions to be easily referenced and distinguished from one another.

Cube Root

The cube root of a number is a special value that, when multiplied by itself three times, gives back the original number. In the expression $y = \sqrt[3]{x}$, $\sqrt[3]{x}$ represents the cube root of $x$.
This operation is different from square roots, which involve multiplying a number by itself only twice.

Here's how cube roots work:

For a positive number $x$, $\sqrt[3]{x}$ is also a positive number. For example, $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$.
The cube root of zero is zero, $\sqrt[3]{0} = 0$, since $0 \times 0 \times 0 = 0$.
For negative numbers, the cube root is negative. For instance, $\sqrt[3]{-8} = -2$ because $-2 \times -2 \times -2 = -8$.

Understanding cube roots is important in solving equations that involve terms raised to the third power and is a fundamental concept in algebra and beyond.

Function Evaluation

Function evaluation involves computing the value of a function for a particular input. Using function notation, evaluating $f(x) = \sqrt[3]{x}$ is as simple as replacing $x$ with the specific value you're interested in.
Here's how you do it:

Take the function expression and plug in the value of $x$ you want to evaluate, like evaluating $f(8)$.
Compute the expression: for $f(x) = \sqrt[3]{x}$, evaluate $f(8) = \sqrt[3]{8}$.
Calculate the result: since $\sqrt[3]{8} = 2$, the value of the function at $x = 8$ is 2, so $f(8) = 2$.

Function evaluation is useful for determining the output of a function for known inputs, which is helpful in problem-solving and understanding functional relationships.

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91影视

Identify the equations that define \(y\) as a function of \(x .\) $$y=\sqrt[3]{x}$$

Short Answer

Step by step solution

Identifying the Function

Function Notation

Example of Function Evaluation

Key Concepts

Function Notation

Cube Root

Function Evaluation

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