Jinsi ya kukokotoa Inverse Function
Inverse Function ni nini?
The inverse function f⁻¹(x) undoes the effect of f(x). If f(a) = b, then f⁻¹(b) = a. Not every function has an inverse — it must be one-to-one (each output has exactly one input).
Fomula
f⁻¹(f(x)) = x and f(f⁻¹(y)) = y; Solve y = f(x) for x in terms of y
- f
- original function
- f⁻¹
- inverse function
- x, y
- variables
Mwongozo wa Hatua kwa Hatua
- 1Replace f(x) with y
- 2Swap x and y
- 3Solve for y — this is f⁻¹(x)
- 4Verify: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Mifano Iliyotatuliwa
Ingizo
f(x) = 2x + 3
Matokeo
f⁻¹(x) = (x−3)/2
Ingizo
f(x) = x³
Matokeo
f⁻¹(x) = x^(1/3) = ∛x
Maswali yanayoulizwa mara kwa mara
Do all functions have inverses?
No, only one-to-one (injective) functions have inverses. The function must pass the horizontal line test.
What is the relationship between f and f⁻¹ graphs?
They are mirror reflections across the line y = x.
What is the domain of f⁻¹?
The domain of f⁻¹ equals the range of f. They swap.
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