Back to Restrict domain for invertibility — Problem 1 · Task Set 46

Exercises: Restrict Domain for Invertibility

Work through each section in order. Show your work where indicated.

Grade 9·20 problems·~30 min·Common Core Math - HS Functions·standard·hsf-bf-b-4d

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Fluency Practice

$f(x) = x^2$ with domain $\mathbb{R}$ has no inverse. If we restrict to $x \geq 0$ , the inverse is $f^{-1}(x) = \sqrt{x}$ . Why does the restriction fix the problem?

On $x \geq 0$ , no two inputs share the same output, so the inverse can map each output back to a unique input.

On $x \geq 0$ , the function becomes linear and therefore invertible.

Restricting to $x \geq 0$ changes the formula to $|x|$ , which is always positive.

Restriction makes the range smaller, which forces the inverse to exist.