Respuesta :

To determine the inverse of the function f(x) = 3x^2 - 15, follow these steps:

1. **Start with the function f(x):**

- f(x) = 3x^2 - 15

2. **Replace f(x) with y:**

- Let y = 3x^2 - 15

3. **Swap x and y:**

- Swap x and y in the equation:

x = 3y^2 - 15

4. **Solve for y to find the inverse:**

- Rearrange the equation to solve for y:

x + 15 = 3y^2

(x + 15) / 3 = y^2

√((x + 15) / 3) = y

Therefore, the inverse function is y = √((x + 15) / 3).

5. **Verify the inverse:**

- To verify that this is the correct inverse, apply the inverse function to the original function:

Replace y with f^(-1)(x):

f^(-1)(x) = √((x + 15) / 3)

Now, replace x in f^(-1)(x) with the original function f(x) = 3x^2 - 15:

f^(-1)(3x^2 - 15) = √(((3x^2 - 15) + 15) / 3)

f^(-1)(3x^2 - 15) = √(3x^2 / 3)

f^(-1)(3x^2 - 15) = √x^2

f^(-1)(3x^2 - 15) = x

This verifies that the found inverse is correct.

Therefore, the inverse of the function f(x) = 3x^2 - 15 is f^(-1)(x) = √((x + 15) / 3).

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