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Find f⁻¹(x) with detailed step-by-step solutions
An inverse function reverses the operation of the original function. If f(a) = b, then f⁻¹(b) = a. The inverse "undoes" what the original function does.
An inverse function f⁻¹(x) reverses the operation of f(x). If f takes input a and produces output b, then f⁻¹ takes input b and produces output a.
No. Only one-to-one functions (functions that pass the horizontal line test) have inverses. Each output must correspond to exactly one input.
Compose the function with its inverse: f(f⁻¹(x)) should equal x, and f⁻¹(f(x)) should also equal x. If both are true, the inverse is correct.
f⁻¹(x) is read as "f inverse of x" and represents the inverse function. Note: this is NOT the same as 1/f(x). The -1 is notation, not an exponent.
Swapping x and y reflects the function across the line y = x, which geometrically represents the inverse relationship. It exchanges inputs and outputs.
Full parabolas fail the horizontal line test. However, if you restrict the domain to one side of the vertex (e.g., x ≥ 0), then the function becomes one-to-one and has an inverse.
Exponential and logarithmic functions are inverses of each other. If f(x) = a^x, then f⁻¹(x) = log_a(x). This is why logarithms "undo" exponents.
The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. The input and output sets are exchanged.
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