Identify which of the following equations represent functions. Select all that apply. y=4x+13 x=5 x^2+y^2=16 y^2=(1)/(3)x-6 y=3x^2-x-1

To determine which of the given equations represent functions, we need to check if each equation defines a unique output (y-value) for every input (x-value). A relation is a function if, for every x-value in its domain, there is exactly one corresponding y-value.<br /><br />1. **\( y = 4x + 13 \)**: This is a linear equation in slope-intercept form \( y = mx + b \), where m and b are constants. For every x-value, there is exactly one y-value. Therefore, this equation represents a function.<br /><br />2. **\( x = 5 \)**: This equation describes a vertical line, which means for the single x-value of 5, there can be multiple y-values. Thus, it does not pass the vertical line test and does not represent a function.<br /><br />3. **\( x^2 + y^2 = 16 \)**: This equation represents a circle with radius 4 centered at the origin. For some x-values, there are two corresponding y-values (one positive and one negative). Therefore, this equation does not represent a function.<br /><br />4. **\( y^2 = \frac{1}{3}x - 6 \)**: Solving for y gives \( y = \pm \sqrt{\frac{1}{3}x - 6} \). For some x-values, there are two possible y-values (positive and negative square roots). Hence, this equation does not represent a function.<br /><br />5. **\( y = 3x^2 - x - 1 \)**: This is a quadratic equation in standard form \( y = ax^2 + bx + c \). For every x-value, there is exactly one y-value. Therefore, this equation represents a function.<br /><br />In summary, the equations that represent functions are:<br />- \( y = 4x + 13 \)<br />- \( y = 3x^2 - x - 1 \)