Functions
General
Grade 12
Question:
Let f(x) = [x - 1] + \{x\}^{[x]}, x \in (1,3), then f^{-1}(x) is -
\frac{x + 1}{2 + \sqrt{x - 1}} \quad x \in (1,2)
\frac{x - 1}{2 - \sqrt{x - 1}} \quad x \in (2,3)
\frac{x - 1}{2 - \sqrt{x - 1}} \quad x \in (0,1)
\frac{x + 1}{2 + \sqrt{x - 1}} \quad x \in (1,2)
Step-by-Step Solution
<div class="solution"><p>f(x)∈(-∞,1]. Apply g on input set (-∞,1]:</p><p>y<-1: g(y)=y²∈(1,∞)</p><p>-1≤y≤1: g(y)=2y+3∈[1,5]</p><p>Union: [1,∞)</p><p><strong>Answer: <span class="math-inline">\([1,\infty)\)</span></strong></p><div class="trap-box"><strong>Trap:</strong> Don't use full domain of g. Input is restricted to range of f.</div><div class="key-concept"><strong>Key Concept:</strong> Range of g∘f = push range of f through g</div></div>
Correct Answer: A