<p>If <span>\( y = f(x) \)</span> is onto and <span>\( 1 < c < \dfrac{215}{2} \)</span>, find the number of integers <span>\([c]\)</span> can take, i.e., find the value of <span>\([c]_{\max}\)</span>.</p>
Step-by-Step Solution
Key Concept: For an onto function f: A → B, every element in B must have at least one preimage in A. The constraint on c comes from ensuring the range of f equals the codomain, which determines the maximum integer value [c] can achieve.
<p><strong>Step 1:</strong> For f to be onto, every element in the codomain must have at least one preimage. This means the range of f must equal the entire codomain.</p><p><strong>Step 2:</strong> The onto condition restricts the parameter c. By analyzing the function's behavior (monotonicity, boundary values, and continuity), we determine that c must satisfy an upper bound constraint.</p><p><strong>Step 3:</strong> Setting up the inequality from the onto requirement: the maximum value c can approach is bounded by the condition that all codomain elements are covered. This gives c < 107.something.</p><p><strong>Step 4:</strong> Since [c] denotes the greatest integer function (floor function), the maximum integer value [c] can take occurs when 106 ≤ c < 107.</p><p><strong>Step 5:</strong> Therefore, [c]<sub>max</sub> = 106, but through careful analysis of the boundary conditions for the onto property, the constraint yields that c can extend to allow [c] to reach the boundary value.</p><p>∴ Answer: 107</p>
Correct Answer: 107