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

Question

Responses

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

y=6x2+5
𝑦
=
6
𝑥
2
+
5
y equals 6 x squared plus 5

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

y=3x+22x−1

Answer 1

To determine which of the given equations has an inverse that is a function, we need to apply the horizontal line test. This test states that if any horizontal line intersects the graph of the function more than once, the function does not have an inverse that is a function.

Let's analyze each option:

1. \( y = 3x^4 + 1 \): The function \( x^4 \) is a polynomial of even degree and is not one-to-one (it has a minimum at \( x = 0 \) and increases on both sides). Therefore, it fails the horizontal line test.

2. \( y = 6x^2 + 5 \): This function is also a polynomial of even degree. The graph is a parabola opening upwards, which means it fails the horizontal line test as well.

3. \( y = \frac{3x^2}{2} - 4 \): Similar to the previous functions, this is also a quadratic function, and it will fail the horizontal line test.

4. \( y = \frac{3x + 2}{2x - 1} \): This is a rational function. To investigate whether it has an inverse that is a function, we can analyze its behavior. This function does have limits and can be one-to-one depending on the intervals we consider.

To summarize:

• The first three options fail the horizontal line test because they are not one-to-one functions.
• The last option, \( y = \frac{3x + 2}{2x - 1} \), is promising because it does not repeat values and has the potential to be one-to-one.

Conclusion: The equation that has an inverse that is a function is \( y = \frac{3x + 2}{2x - 1} \).