Answered

Join IDNLearn.com today and start getting the answers you've been searching for. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.

Determine if the given equations are functions or not functions:

[tex]\[ x = 3 \][/tex]

[tex]\[ x^2 + y^2 = 81 \][/tex]

[tex]\[ y = -x + 11 \][/tex]

[tex]\[ y = -5x - 12 \][/tex]

[tex]\[ y = 2x^2 - 6x + 4 \][/tex]

[tex]\[ y = -7 \][/tex]

Not Functions:

[tex]\[ x = 3 \][/tex]

[tex]\[ x^2 + y^2 = 81 \][/tex]

Sagot :

GinnyAnsweridn
To determine whether the given equations are functions, we need to recall the definition of a function. A relation is defined as a function if each input (x-value) maps to exactly one output (y-value). Let's scrutinize each equation step-by-step:

1. Equation: [tex]\( x = 3 \)[/tex]
- This equation implies that [tex]\( x \)[/tex] is always 3 regardless of the value of [tex]\( y \)[/tex].
- For a function, each value of [tex]\( x \)[/tex] should produce a unique [tex]\( y \)[/tex] value.
- Here, [tex]\( x \)[/tex] is fixed and can correspond to many different [tex]\( y \)[/tex] values.
- Conclusion: This is not a function.

2. Equation: [tex]\( x^2 + y^2 = 81 \)[/tex]
- This is the equation of a circle with radius 9 centered at the origin.
- For a given [tex]\( x \)[/tex], there can be two [tex]\( y \)[/tex] values (one positive and one negative), except at the points where [tex]\( x = \pm 9 \)[/tex].
- For a function, a unique [tex]\( y \)[/tex] value must be paired with each [tex]\( x \)[/tex] value, which is not the case here.
- Conclusion: This is not a function.

3. Equation: [tex]\( y = -x + 11 \)[/tex]
- This is a linear equation of the form [tex]\( y = mx + b \)[/tex].
- For each [tex]\( x \)[/tex], there is a unique [tex]\( y \)[/tex] value.
- Conclusion: This is a function.

4. Equation: [tex]\( y = -5x - 12 \)[/tex]
- Again, this is a linear equation.
- Each [tex]\( x \)[/tex] value gives exactly one [tex]\( y \)[/tex] value.
- Conclusion: This is a function.

5. Equation: [tex]\( y = 2x^2 - 6x + 4 \)[/tex]
- This is a quadratic equation, which represents a parabola.
- For each [tex]\( x \)[/tex] value, there is a unique [tex]\( y \)[/tex] value.
- Conclusion: This is a function.

6. Equation: [tex]\( y = -7 \)[/tex]
- This is a constant function, where [tex]\( y \)[/tex] is always -7 regardless of the value of [tex]\( x \)[/tex].
- Each [tex]\( x \)[/tex] value maps to the same [tex]\( y \)[/tex] value.
- Conclusion: This is a function.

Therefore, among the six equations, the ones that are not functions are:

1. [tex]\( x = 3 \)[/tex]
2. [tex]\( x^2 + y^2 = 81 \)[/tex]

Thus, there are 2 equations that are not functions.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Trust IDNLearn.com for all your queries. We appreciate your visit and hope to assist you again soon.