Relations & Functions
Inverse Functions
Grade 12
Question:
<p>Let \(f : \mathbb{R} \to \mathbb{R}\) is defined by \(f(x) = \begin{cases} (x+1)^3 & ; x \leq 1 \\ \ln x + (b^2 - 3b + 10) & ; x > 1 \end{cases}\). If \(f(x)\) is invertible, then the set of all values of \(b\) is:</p>
<p>(a) \(\{1, 2\}\)</p>
<p>(b) \(\emptyset\)</p>
<p>(c) \(\{2, 5\}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For a piecewise function to be invertible, it must be both one-one and onto. Check continuity at \(x=1\) and monotonicity on each piece. The value at \(x=1\) from the first piece is \(8\); ensure the second piece doesn't produce the same values elsewhere.
<p>Solution not provided in source text.</p>
Correct Answer: a