To determine which function shares no real solutions with the quadratic function represented by the parabola that does not touch the x-axis, we need to analyze the options:
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y = -1/2x + 6: This is a linear function. A linear function can intersect a quadratic function at one or two points, depending on its slope and y-intercept. Thus, it might have real solutions with the quadratic function.
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y = x^2: This is the same type of function as the parabola, and it touches the x-axis at the vertex if it's the same quadratic function. Therefore, it could share solutions.
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y = x + 1: This is also a linear function. Like option 1, it can intersect a quadratic function based on its slope. Therefore, it can have real solutions.
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y = 1/2x - 1: Again, this is a linear function. Similar to the other linear functions, it can also intersect a quadratic function, depending on its slope and intercept.
Given that the parabola does not touch the x-axis, it means it has no real roots, and it opens either upwards or downwards without touching or crossing the x-axis. If a linear function can intersect this parabola, it would create a real solution.
However, looking more closely, the options of linear functions (a, c, d) inherently can intersect the quadratic function at one or more points unless their slope and y-intercept specifically position them such that they lie completely above or below the range of the quadratic function.
Given that the quadratic does not touch the x-axis, we are looking for a function that is entirely beneath or above the range of the quadratic.
Since we're specifically looking for a function that shares no real solutions, the function that does not intersect the parabola is most likely a linear function that lies entirely above the maximum point of the parabola.
If the quadratic opens upwards (typical case), a potential candidate could be a line with a high y-intercept like y = -1/2x + 6, considering that it provides a positive value for y across many x-values.
However, since the function is a linear one and would vary downwards or across \(x\), it can't guarantee zero intersections without knowing the precise position of the quadratic.
So, assuming the parabola has a minimum point strictly above zero on the y-axis, a function like y = 1/2x - 1 may potentially offer no real solutions if its slope doesn’t allow it to ascend.
In conclusion, if constrained to deduce without viewing the graph, choice d. y = 1/2x - 1 might be argued to hold no real solutions specifically beneath a certain threshold aligned to the graph of the quadratic itself.
Thus, the safest choice for a function that shares no real solutions would be: d. y = 1/2x - 1.