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Flip to reveal answersHow do you find f⁻¹ algebraically?
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All 8 Flashcards — Finding the inverse
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Question
How do you find f⁻¹ algebraically?
Answer
Write y = f(x), swap x and y, solve for y. (Geometrically, reflect in y = x.)
Question
How do you invert a rational function?
Answer
Swap x and y, multiply up to clear the fraction, gather the y-terms, factor out y, then divide.
Question
Find the inverse of f(x) = 5x − 2.
Answer
x = 5y − 2 ⇒ y = (x + 2)/5, so f⁻¹(x) = (x + 2)/5.
Question
What composition check confirms an inverse?
Answer
f(f⁻¹(x)) = x (and f⁻¹(f(x)) = x) — they undo each other.
Question
How do the domain and range of f⁻¹ relate to f?
Answer
They swap: domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.
Question
Why does x² need a restricted domain to have an inverse?
Answer
x² isn't one-to-one over all x; restricting to x ≥ 0 makes it invertible, giving f⁻¹(x) = √x.
Question
Inverse of f(x) = (2x + 1)/(x − 3)?
Answer
Swap, multiply up, gather y: f⁻¹(x) = (3x + 1)/(x − 2).
Question
Is f⁻¹ the same as 1/f?
Answer
No — f⁻¹ is the inverse function; 1/f is the reciprocal.
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Topic 2.5 hub
Composite & inverse functions
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