question the graph of which of the following in the xy - plane does not…

Question

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

$y = \\sqrt{x}+11$
$y = \\sqrt{x + 11}$
$y = 11 - \\sqrt{x}$
$y = \\sqrt{11 - x}$

Solution Steps

Understand the question
question
the graph of which of the following in the xy - plane does not intersect the line $y = 10$?
- $y = \\sqrt{x}+11$
- $y = \\sqrt{x + 11}$
- $y = 11 - \\sqrt{x}$
- $y = \\sqrt{11 - x}$
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 $.
Final answer
$ y = \sqrt{x}+11 $ (the first option)

Answer

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 $.

Answer

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

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Question Analysis

Subject mathematics

Sub Subject calculus

Education Level high school

Difficulty unspecified

Question Type multiple choice

Multi Question No

Question Count 1

Analysis Status completed

Analyzed At 2026-02-04T16:23:31

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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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