Question

Inverses and Functions Quick Check
2 of 52 of 5 Items


Question
Which of the following equations has an inverse that is a function?(1 point)
Responses

y=6x2+5
y equals 6 x squared plus 5

y=3x+22x−1
y equals Start Fraction 3 x plus 2 over 2 x minus 1 End Fraction

y=3x4+1
y equals 3 x superscript 4 baseline plus 1

y=3x22−4
y equals Start Fraction 3 x squared over 2 End Fraction minus 4
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Answer 1

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, meaning that for every horizontal line drawn, it intersects the graph of the function at most once.

Let's analyze each option:

1. **y = 6x² + 5**: This is a quadratic function, which opens upwards. Since it is a parabola, it does not pass the horizontal line test (a horizontal line can intersect the parabola at two points). Therefore, this function does not have an inverse that is a function.

2. **y = (3x + 2) / (2x - 1)**: This is a rational function. To find out if it is one-to-one, we can check its derivative. The derivative is positive for all x in the domain, which implies it is strictly increasing. Therefore, it passes the horizontal line test and its inverse is a function.

3. **y = 3x⁴ + 1**: This is also a polynomial function, specifically of degree four. It has a local minimum and can take on the same value at different points (for example, there are two values of x that yield the same y for negative and positive x values). Thus, it fails the horizontal line test and does not have an inverse that is a function.

4. **y = (3x²) / 2 - 4**: This is a quadratic function (specifically, a downward-opening parabola when organized properly). Like the first equation, it does not pass the horizontal line test. Therefore, it does not have an inverse that is a function.

Based on this analysis, the only equation that has an inverse that is a function is:

**y = (3x + 2) / (2x - 1)**.