How can I simplify the equation y = 5x² + 10 to find its inverse?

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To find the inverse of the equation y = 5x² + 10, we follow several steps to simplify it and express x in terms of y.

Step 1: Swap the Variables

The first step in finding the inverse is to switch x and y in the equation. This gives us:

x = 5y² + 10

Step 2: Isolate the Quadratic Term

Next, we isolate the y² term. To do this, we subtract 10 from both sides of the equation:

x - 10 = 5y²

Then, divide both sides by 5 to further isolate y²:

(x - 10) / 5 = y²

Step 3: Solve for y

Now, we take the square root of both sides to solve for y.

y = ±√((x - 10) / 5)

Here, the ± sign indicates that there are two possible values for y for each value of x, which corresponds to the symmetrical properties of the quadratic function.

Step 4: Conclusion

Thus, the inverse of the original equation y = 5x² + 10 can be expressed as:

y = ±√((x - 10) / 5)

Bear in mind that since the original function represents a parabola, specifying the range for which the inverse is valid (either the positive or the negative square root) will be essential depending on the context in which you use the inverse function.

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