QuestionDetermine if the following equations define $y$ as a function of $x$ : 28. $y=x^{3}$ , 31. $y^{2}=4-x^{2}$ , $y=|x-3|$ .

Studdy Solution

STEP 1

Assumptions1. We are given three equations $y=x^{3}$ , $y^{}=4-x^{}$ , and $y=|x-3|$ . We need to determine whether each equation defines $y$ as a function of $x$

STEP 2

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In other words, for every $x$ value, there should be exactly one corresponding $y$ value.

STEP 3

Let's start with the first equation $y=x^{3}$ . For every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.

STEP 4

Now let's consider the second equation $y^{2}=4-x^{2}$ . To see if $y$ is a function of $x$ , we can solve for $y$ .

STEP 5

olving for $y$ in the equation $y^{2}=4-x^{2}$ gives us $y=\sqrt{4-x^{2}}$ and $y=-\sqrt{4-x^{2}}$

STEP 6

This means that for some values of $x$ , there are two possible values of $y$ . Therefore, $y$ is not a function of $x$ in this equation.

STEP 7

Finally, let's consider the third equation $y=|x-3|$ . The absolute value function always gives a non-negative output for any input. Therefore, for every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.

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QuestionDetermine if the following equations define $y$ as a function of $x$ : 28. $y=x^{3}$ , 31. $y^{2}=4-x^{2}$ , $y=|x-3|$ .

Studdy Solution

STEP 1

Assumptions1. We are given three equations $y=x^{3}$ , $y^{}=4-x^{}$ , and $y=|x-3|$ . We need to determine whether each equation defines $y$ as a function of $x$

STEP 2

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In other words, for every $x$ value, there should be exactly one corresponding $y$ value.

STEP 3

Let's start with the first equation $y=x^{3}$ . For every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.

STEP 4

Now let's consider the second equation $y^{2}=4-x^{2}$ . To see if $y$ is a function of $x$ , we can solve for $y$ .

STEP 5

olving for $y$ in the equation $y^{2}=4-x^{2}$ gives us $y=\sqrt{4-x^{2}}$ and $y=-\sqrt{4-x^{2}}$

STEP 6

This means that for some values of $x$ , there are two possible values of $y$ . Therefore, $y$ is not a function of $x$ in this equation.

STEP 7

Finally, let's consider the third equation $y=|x-3|$ . The absolute value function always gives a non-negative output for any input. Therefore, for every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.

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QuestionDetermine if the following equations define yyy as a function of xxx: 28. y=x3y=x^{3}y=x3, 31. y2=4−x2y^{2}=4-x^{2}y2=4−x2, y=∣x−3∣y=|x-3|y=∣x−3∣.

Studdy Solution

STEP 1

Assumptions1. We are given three equations y=x3y=x^{3}y=x3, y=4−xy^{}=4-x^{}y=4−x, and y=∣x−3∣y=|x-3|y=∣x−3∣ . We need to determine whether each equation defines yyy as a function of xxx

STEP 2

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In other words, for every xxx value, there should be exactly one corresponding yyy value.

STEP 3

Let's start with the first equation y=x3y=x^{3}y=x3. For every value of xxx, there is exactly one corresponding value of yyy. So, yyy is a function of xxx in this equation.

STEP 4

Now let's consider the second equation y2=4−x2y^{2}=4-x^{2}y2=4−x2. To see if yyy is a function of xxx, we can solve for yyy.

STEP 5

olving for yyy in the equation y2=4−x2y^{2}=4-x^{2}y2=4−x2 gives usy=4−x2y=\sqrt{4-x^{2}}y=4−x2​andy=−4−x2y=-\sqrt{4-x^{2}}y=−4−x2​

STEP 6

This means that for some values of xxx, there are two possible values of yyy. Therefore, yyy is not a function of xxx in this equation.

STEP 7

Finally, let's consider the third equation y=∣x−3∣y=|x-3|y=∣x−3∣. The absolute value function always gives a non-negative output for any input. Therefore, for every value of xxx, there is exactly one corresponding value of yyy. So, yyy is a function of xxx in this equation.

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QuestionDetermine if the following equations define $y$ as a function of $x$ : 28. $y=x^{3}$ , 31. $y^{2}=4-x^{2}$ , $y=|x-3|$ .

Assumptions1. We are given three equations $y=x^{3}$ , $y^{}=4-x^{}$ , and $y=|x-3|$ . We need to determine whether each equation defines $y$ as a function of $x$

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In other words, for every $x$ value, there should be exactly one corresponding $y$ value.

Let's start with the first equation $y=x^{3}$ . For every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.

Now let's consider the second equation $y^{2}=4-x^{2}$ . To see if $y$ is a function of $x$ , we can solve for $y$ .

olving for $y$ in the equation $y^{2}=4-x^{2}$ gives us $y=\sqrt{4-x^{2}}$ and $y=-\sqrt{4-x^{2}}$

This means that for some values of $x$ , there are two possible values of $y$ . Therefore, $y$ is not a function of $x$ in this equation.

Finally, let's consider the third equation $y=|x-3|$ . The absolute value function always gives a non-negative output for any input. Therefore, for every value of $x$ , there is exactly one corresponding value of $y$ . So, $y$ is a function of $x$ in this equation.