The inverse function f⁻¹(x) = a means that f(a) = x, so to find f⁻¹(7), you need to find the input value 'a' that produces an output of 7 when passed through the original function f. For the function f(x) = √(3x - 8), solving f(a) = 7 by squaring both sides gives 3a - 8 = 49, so 3a = 57, and a = 19. Therefore, f⁻¹(7) = 19. The key is understanding that inverse functions reverse the input-output relationship rather than just memorizing algebraic steps.
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“Why This Inverse Shortcut Works ⚡”Añadido:
This inverse function shortcut is easy unless you know the core concept behind it.
Most students do this correctly, but for the wrong reason.
They square both the sides without understanding what f inverse of 7 actually means. It's asking which input gives an output of 7.
Here is the real concept. If inverse at 7 equals a, then the function at a must return 7.
To solve this radical function, we need to take square on both the sides and then solve for a. We got a equal to 19 and therefore f inverse of 7 is equal to 19. Now the algebra actually makes sense. Don't just memorize the steps, know the actual reasoning behind the concept.
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