c) Given the function fሺxሻ=2xଶ െ 3, find its inverse function fିଵሺxሻ.
Question
c) Given the function fሺxሻ=2xଶ െ 3, find its inverse function fିଵሺxሻ.
Solution
To find the inverse of the function f(x) = 2x² - 3, we follow these steps:
- Replace f(x) with y. This gives us y = 2x² - 3.
- Swap x and y. This gives us x = 2y² - 3.
- Solve for y.
First, add 3 to both sides to get x + 3 = 2y².
Then, divide both sides by 2 to get (x + 3) / 2 = y².
Finally, take the square root of both sides to solve for y. Remember to consider both the positive and negative roots. This gives us y = sqrt((x + 3) / 2) and y = -sqrt((x + 3) / 2).
So, the inverse function f⁻¹(x) is f⁻¹(x) = sqrt((x + 3) / 2) and f⁻¹(x) = -sqrt((x + 3) / 2).
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