To determine which of the given equations has an inverse that is a function, we need to check if the function is one-to-one. A function is one-to-one if it passes the horizontal line test; that is, no horizontal line intersects the graph of the function more than once.
Let's analyze each function:
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\( y = \frac{3x^2}{2} - 4 \)
- This is a quadratic function (parabola) which opens upwards. It is not one-to-one because horizontal lines will intersect the graph at two points (except at the vertex).
- Does not have an inverse that is a function.
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\( y = 3x^4 + 1 \)
- This is a polynomial function of degree 4, which also leads to a shape that is not one-to-one (the ends go upwards, and the middle can turn back downwards).
- Does not have an inverse that is a function.
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\( y = 6x^2 + 5 \)
- This is another quadratic function (also a parabola) that opens upwards. Similar to the first case, it is not one-to-one.
- Does not have an inverse that is a function.
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\( y = \frac{3x + 2}{2x - 1} \)
- This is a rational function. To determine if it is one-to-one, we can analyze its behavior. A rational function can be one-to-one if it doesn't repeat values.
- The first derivative test can be performed or simply checking if it has horizontal asymptotes would usually help confirm: the function does not oscillate or turn back on itself, suggesting it might indeed be one-to-one throughout its domain.
- May have an inverse that is a function.
Based on this analysis, the fourth function \( y = \frac{3x + 2}{2x - 1} \) is the only one that is likely to have an inverse that is a function.
2 answers