To determine which of the given equations represent functions, we need to check if each equation passes the vertical line test. Specifically, for an equation to be a function, each value of [tex]\( x \)[/tex] must correspond to exactly one value of [tex]\( y \)[/tex].
Let's analyze each equation in detail:
1. Equation: [tex]\( y = 4x + 13 \)[/tex]
- This is a linear equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants.
- For any given [tex]\( x \)[/tex], there is exactly one corresponding [tex]\( y \)[/tex].
- Conclusion: This is a function.
2. Equation: [tex]\( x = 5 \)[/tex]
- This represents a vertical line at [tex]\( x = 5 \)[/tex].
- A vertical line does not satisfy the definition of a function, as it has an infinite number of [tex]\( y \)[/tex]-values for a single [tex]\( x \)[/tex]-value.
- Conclusion: This is not a function.
3. Equation: [tex]\( x^2 + y^2 = 16 \)[/tex]
- This is the equation of a circle with a radius of 4 and centered at the origin (0, 0).
- For certain values of [tex]\( x \)[/tex], there will be two corresponding [tex]\( y \)[/tex]-values (e.g., at [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] can be [tex]\( 4 \)[/tex] or [tex]\( -4 \)[/tex]).
- Conclusion: This is not a function.
4. Equation: [tex]\( y^2 = \frac{1}{3}x - 6 \)[/tex]
- Solving for [tex]\( y \)[/tex], we get [tex]\( y = \pm \sqrt{\frac{1}{3}x - 6} \)[/tex].
- For certain values of [tex]\( x \)[/tex], there will be two corresponding [tex]\( y \)[/tex]-values (one positive and one negative).
- Conclusion: This is not a function.
5. Equation: [tex]\( y = 3x^2 - x - 1 \)[/tex]
- This is a quadratic equation in [tex]\( x \)[/tex] with the general form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- For any given [tex]\( x \)[/tex], there is exactly one corresponding [tex]\( y \)[/tex].
- Conclusion: This is a function.
Based on this analysis, the equations that represent functions are:
1. [tex]\( y = 4x + 13 \)[/tex]
5. [tex]\( y = 3x^2 - x - 1 \)[/tex]
Therefore, the indices of the equations that are functions are [tex]\([1, 5]\)[/tex].
