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).
