Question

Which of the following equations has an inverse that is a function?(1 point) Responses y=3x22−4 y equals Start Fraction 3 x squared over 2 End Fraction minus 4 y=3x4+1 y equals 3 x superscript 4 baseline plus 1 y=3x+22x−1 y equals Start Fraction 3 x plus 2 over 2 x minus 1 End Fraction y=6x2+5 y equals 6 x squared plus 5

Answer 1

To determine which of the given equations has an inverse that is a function, we need to check whether each equation is one-to-one. A function is one-to-one if it passes the Horizontal Line Test, meaning that any horizontal line drawn through the graph of the function intersects the graph at most once.

Let's analyze each equation:

1. \( y = \frac{3x^2}{2} - 4 \):

   • This is a quadratic function (parabola) that opens upwards. It will fail the Horizontal Line Test because horizontal lines will intersect the graph at two points (unless the vertex is the only point of interest).

2. \( y = 3x^4 + 1 \):

   • This is also a polynomial function (quartic) and will not be one-to-one, as it will also produce multiple outputs for certain inputs (like other even-degree polynomials).

3. \( y = \frac{3x + 2}{2x - 1} \):

   • This is a rational function. To check for one-to-oneness, we analyze the function's behavior. As it's not a polynomial, we find that there may be values of \(x\) that produce the same \(y\), but we need to analyze it further. However, rational functions can sometimes be one-to-one depending on their specific form.

4. \( y = 6x^2 + 5 \):

   • Like the first equation, this is another quadratic function that opens upwards. It also fails the Horizontal Line Test.

From the analysis, the first, second, and fourth equations are not one-to-one. The third equation (\( y = \frac{3x + 2}{2x - 1} \)) needs further exploration, but it can potentially be one-to-one depending on its critical points.

The only rational function here could potentially have an inverse function (depending on the domain restrictions), but given the ambiguity, when asked to find a clear example inclusive of one-to-one functions, the conclusion is that none of the more standard functions here are guaranteed to have inverses that are also functions without introducing specific domain considerations.

Based on this analysis, the likely candidate for an inverse that may behave according to function rules is:

\( y = \frac{3x + 2}{2x - 1} \)

Final answer: \( y = \frac{3x + 2}{2x - 1} \) might have an inverse that is a function, depending on defined domain conditions.