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题目

解题步骤

1. Understand the question

   question
   the graph of which of the following in the xy - plane does not intersect the line $y = 10$?

   • $y = \\sqrt{x}+11$
   • $y = \\sqrt{x + 11}$
   • $y = 11 - \\sqrt{x}$
   • $y = \\sqrt{11 - x}$
2. Explanation

   Step1: Analyze \( y = \sqrt{x}+11 \)

   Set \( y = 10 \), so \( 10=\sqrt{x}+11 \). Then \( \sqrt{x}=10 - 11=-1 \). But \( \sqrt{x}\geq0 \) for real \( x \), no solution? Wait, no, wait: \( \sqrt{x}=-1 \) has no real solution, but wait, let's check other options. Wait, maybe I made a mistake. Wait, let's check each function:

   Step2: Analyze \( y=\sqrt{x + 11} \)

   Set \( y = 10 \), so \( 10=\sqrt{x + 11} \). Square both sides: \( 100=x + 11 \), so \( x=89 \). Valid, since \( x+11=100\geq0 \). So this intersects \( y = 10 \).

   Step3: Analyze \( y = 11-\sqrt{x} \)

   Set \( y = 10 \), so \( 10=11-\sqrt{x} \). Then \( \sqrt{x}=11 - 10 = 1 \), so \( x = 1 \). Valid, since \( \sqrt{1}=1 \). So this intersects \( y = 10 \).

   Step4: Analyze \( y=\sqrt{11 - x} \)

   Set \( y = 10 \), so \( 10=\sqrt{11 - x} \). Square both sides: \( 100=11 - x \), so \( x=11 - 100=-89 \). Then check the domain: \( 11 - x\geq0\Rightarrow x\leq11 \). \( x=-89 \) is valid, so this intersects \( y = 10 \).

   Wait, wait, earlier step1: \( y=\sqrt{x}+11 \), set \( y = 10 \): \( \sqrt{x}=10 - 11=-1 \). But \( \sqrt{x}\geq0 \), so no solution. So this graph does not intersect \( y = 10 \).

3. Final answer

   \( y = \sqrt{x}+11 \) (the first option)

答案

Question Analysis

OCR Text

question
the graph of which of the following in the xy - plane does not intersect the line $y = 10$?
- $y = \\sqrt{x}+11$
- $y = \\sqrt{x + 11}$
- $y = 11 - \\sqrt{x}$
- $y = \\sqrt{11 - x}$

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