Math / Algebra

QuestionDetermine whether the equation defines $y$ as a function of $x$ . (See Example 9.) $3|x|+y=0$ is a function is not a function

Studdy Solution

STEP 1

1. A function is defined as a relation where each input (in this case, each $x$ ) has exactly one output ( $y$ ).
2. We need to determine if for every $x$ , there is a unique $y$ that satisfies the equation.

STEP 2

1. Isolate $y$ in the equation.
2. Analyze the expression for $y$ to determine if it is unique for each $x$ .
3. Conclude whether $y$ is a function of $x$ .

STEP 3

Start with the given equation:
$3|x| + y = 0$
Isolate $y$ by subtracting $3|x|$ from both sides:
$y = -3|x|$

STEP 4

Analyze the expression $y = -3|x|$ .
The absolute value function $|x|$ is unique for each $x$ , meaning for each $x$ , there is exactly one value of $|x|$ .
Since $y$ is expressed as $-3$ times the absolute value of $x$ , for each $x$ , there is exactly one corresponding $y$ .

STEP 5

Conclude whether $y$ is a function of $x$ .
Since for each $x$ , there is exactly one $y$ , the equation $y = -3|x|$ defines $y$ as a function of $x$ .
The equation defines $y$ as a function of $x$ .
The equation is a function.

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QuestionDetermine whether the equation defines $y$ as a function of $x$ . (See Example 9.) $3|x|+y=0$ is a function is not a function

Studdy Solution

STEP 1

1. A function is defined as a relation where each input (in this case, each $x$ ) has exactly one output ( $y$ ).
2. We need to determine if for every $x$ , there is a unique $y$ that satisfies the equation.

STEP 2

1. Isolate $y$ in the equation.
2. Analyze the expression for $y$ to determine if it is unique for each $x$ .
3. Conclude whether $y$ is a function of $x$ .

STEP 3

Start with the given equation:
$3|x| + y = 0$
Isolate $y$ by subtracting $3|x|$ from both sides:
$y = -3|x|$

STEP 4

Analyze the expression $y = -3|x|$ .
The absolute value function $|x|$ is unique for each $x$ , meaning for each $x$ , there is exactly one value of $|x|$ .
Since $y$ is expressed as $-3$ times the absolute value of $x$ , for each $x$ , there is exactly one corresponding $y$ .

STEP 5

Conclude whether $y$ is a function of $x$ .
Since for each $x$ , there is exactly one $y$ , the equation $y = -3|x|$ defines $y$ as a function of $x$ .
The equation defines $y$ as a function of $x$ .
The equation is a function.

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QuestionDetermine whether the equation defines yyy as a function of xxx. (See Example 9.) 3∣x∣+y=03|x|+y=03∣x∣+y=0 is a function is not a function

Studdy Solution

STEP 1

1. A function is defined as a relation where each input (in this case, each x x x) has exactly one output (y y y).2. We need to determine if for every x x x, there is a unique y y y that satisfies the equation.

STEP 2

1. Isolate y y y in the equation.2. Analyze the expression for y y y to determine if it is unique for each x x x.3. Conclude whether y y y is a function of x x x.

STEP 3

Start with the given equation:3∣x∣+y=0 3|x| + y = 0 3∣x∣+y=0Isolate y y y by subtracting 3∣x∣ 3|x| 3∣x∣ from both sides:y=−3∣x∣ y = -3|x| y=−3∣x∣

STEP 4

STEP 5

Conclude whether y y y is a function of x x x.Since for each x x x, there is exactly one y y y, the equation y=−3∣x∣ y = -3|x| y=−3∣x∣ defines y y y as a function of x x x.The equation defines y y y as a function of x x x. The equation is a function.

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QuestionDetermine whether the equation defines $y$ as a function of $x$ . (See Example 9.) $3|x|+y=0$ is a function is not a function

1. A function is defined as a relation where each input (in this case, each $x$ ) has exactly one output ( $y$ ).
2. We need to determine if for every $x$ , there is a unique $y$ that satisfies the equation.

1. Isolate $y$ in the equation.
2. Analyze the expression for $y$ to determine if it is unique for each $x$ .
3. Conclude whether $y$ is a function of $x$ .

Start with the given equation:
$3|x| + y = 0$
Isolate $y$ by subtracting $3|x|$ from both sides:
$y = -3|x|$

Conclude whether $y$ is a function of $x$ .
Since for each $x$ , there is exactly one $y$ , the equation $y = -3|x|$ defines $y$ as a function of $x$ .
The equation defines $y$ as a function of $x$ .
The equation is a function.