Answer :
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.
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.
