Answer :
To identify which of the given equations represent functions, we need to understand the definition of a function. A function is a relation in which each input (typically an [tex]\( x \)[/tex]-value) is associated with exactly one output (typically a [tex]\( y \)[/tex]-value).
Let's analyze each equation step by step:
### Equation 1: [tex]\( y = 4x + 13 \)[/tex]
- This equation expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex].
- For each [tex]\( x \)[/tex]-value, there is exactly one corresponding [tex]\( y \)[/tex]-value.
- Therefore, this equation represents a function.
### Equation 2: [tex]\( x = 5 \)[/tex]
- This equation states that [tex]\( x \)[/tex] is always equal to 5, regardless of the value of [tex]\( y \)[/tex].
- It does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]; instead, it represents a vertical line in the [tex]\( xy \)[/tex]-plane.
- Therefore, this equation does not represent a function.
### Equation 3: [tex]\( x^2 \cdot y^2 = 16 \)[/tex]
- To determine if this represents a function, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
[tex]\[ y^2 = \frac{16}{x^2} \][/tex]
[tex]\[ y = \pm \frac{4}{|x|} \][/tex]
- For a given [tex]\( x \)[/tex], there can be two corresponding [tex]\( y \)[/tex]-values ([tex]\( y = \frac{4}{|x|} \)[/tex] and [tex]\( y = -\frac{4}{|x|} \)[/tex]).
- Therefore, this equation does not represent a function.
### Equation 4: [tex]\( y^2 = \frac{1}{3}x - 6 \)[/tex]
- Solve this equation for [tex]\( y \)[/tex].
[tex]\[ y = \pm \sqrt{\frac{1}{3}x - 6} \][/tex]
- For a given [tex]\( x \)[/tex], there can be two corresponding [tex]\( y \)[/tex]-values ([tex]\( y = \sqrt{\frac{1}{3}x - 6} \)[/tex] and [tex]\( y = -\sqrt{\frac{1}{3}x - 6} \)[/tex]).
- Therefore, this equation does not represent a function.
### Equation 5: [tex]\( y = 3x^2 - x - 1 \)[/tex]
- This equation expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex].
- For each [tex]\( x \)[/tex]-value, there is exactly one corresponding [tex]\( y \)[/tex]-value.
- Therefore, this equation represents a function.
Summarizing, the equations that represent functions are:
1. [tex]\( y = 4x + 13 \)[/tex]
2. [tex]\( y = 3x^2 - x - 1 \)[/tex]
Thus, the equations that represent functions are:
[tex]\[ y = 4x + 13 \][/tex]
[tex]\[ y = 3x^2 - x - 1 \][/tex]
Let's analyze each equation step by step:
### Equation 1: [tex]\( y = 4x + 13 \)[/tex]
- This equation expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex].
- For each [tex]\( x \)[/tex]-value, there is exactly one corresponding [tex]\( y \)[/tex]-value.
- Therefore, this equation represents a function.
### Equation 2: [tex]\( x = 5 \)[/tex]
- This equation states that [tex]\( x \)[/tex] is always equal to 5, regardless of the value of [tex]\( y \)[/tex].
- It does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]; instead, it represents a vertical line in the [tex]\( xy \)[/tex]-plane.
- Therefore, this equation does not represent a function.
### Equation 3: [tex]\( x^2 \cdot y^2 = 16 \)[/tex]
- To determine if this represents a function, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
[tex]\[ y^2 = \frac{16}{x^2} \][/tex]
[tex]\[ y = \pm \frac{4}{|x|} \][/tex]
- For a given [tex]\( x \)[/tex], there can be two corresponding [tex]\( y \)[/tex]-values ([tex]\( y = \frac{4}{|x|} \)[/tex] and [tex]\( y = -\frac{4}{|x|} \)[/tex]).
- Therefore, this equation does not represent a function.
### Equation 4: [tex]\( y^2 = \frac{1}{3}x - 6 \)[/tex]
- Solve this equation for [tex]\( y \)[/tex].
[tex]\[ y = \pm \sqrt{\frac{1}{3}x - 6} \][/tex]
- For a given [tex]\( x \)[/tex], there can be two corresponding [tex]\( y \)[/tex]-values ([tex]\( y = \sqrt{\frac{1}{3}x - 6} \)[/tex] and [tex]\( y = -\sqrt{\frac{1}{3}x - 6} \)[/tex]).
- Therefore, this equation does not represent a function.
### Equation 5: [tex]\( y = 3x^2 - x - 1 \)[/tex]
- This equation expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex].
- For each [tex]\( x \)[/tex]-value, there is exactly one corresponding [tex]\( y \)[/tex]-value.
- Therefore, this equation represents a function.
Summarizing, the equations that represent functions are:
1. [tex]\( y = 4x + 13 \)[/tex]
2. [tex]\( y = 3x^2 - x - 1 \)[/tex]
Thus, the equations that represent functions are:
[tex]\[ y = 4x + 13 \][/tex]
[tex]\[ y = 3x^2 - x - 1 \][/tex]
