Question:
Let \( g(x) = \max(5 - x,\ x + 2) \). The smallest possible value of \( g(x) \) is
Let \( g(x) = \max(5 - x,\ x + 2) \). The smallest possible value of \( g(x) \) is
Show Hint
Where do the lines \( 5 - x \) and \( x + 2 \) meet? The maximum of two such lines is lowest at that meeting point. Compare the value you get there with the options.
Updated On: Nov 7, 2026
- \( 4.0 \)
- \( 4.5 \)
- \( 1.5 \)
- None of the above
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The function \(g(x) = \max(5 - x,\ x + 2)\) takes, at every \(x\), the larger of the two values \(5 - x\) and \(x + 2\). We need the smallest value this larger quantity can ever reach.
The line \(5 - x\) goes down as \(x\) grows, and the line \(x + 2\) goes up as \(x\) grows. So one line is high where the other is low, and the maximum of the two will look like the letter V.
Step 2: Find where the two lines cross:
The V has its lowest point exactly where the two lines meet, so we solve \(5 - x = x + 2\).
\[ 5 - 2 = x + x \Rightarrow 3 = 2x \Rightarrow x = 1.5 \]
At this point both expressions are equal: \(5 - 1.5 = 3.5\) and \(1.5 + 2 = 3.5\).
Step 3: Write g(x) as two pieces:
For \(x \leq 1.5\) we have \(5 - x \geq x + 2\), so \(g(x) = 5 - x\). This piece keeps falling as \(x\) rises, and its smallest value on this side is at \(x = 1.5\), namely \(3.5\).
For \(x \geq 1.5\) we have \(x + 2 \geq 5 - x\), so \(g(x) = x + 2\). This piece keeps rising as \(x\) rises, so its smallest value on this side is also at \(x = 1.5\), namely \(3.5\).
Hence the smallest possible value of \(g(x)\) is \(3.5\).
Step 4: Check option 1:
The value \(4.0\) can be reached by \(g\), but it is not the smallest, because \(g\) comes down to \(3.5\). So option 1 is wrong.
Step 5: Check option 2:
The value \(4.5\) is also reached (for example at \(x = 0.5\), where \(g = 4.5\)), but again it is larger than \(3.5\). So option 2 is wrong.
Step 6: Check option 3:
The number \(1.5\) is the place where the minimum occurs, not the minimum value. Also \(g(x)\) is never below \(3.5\), so it can never be \(1.5\). This is a trap option. So option 3 is wrong.
Step 7: Check option 4:
The true minimum \(3.5\) is not among the first three options, so "None of the above" is correct.
Final Answer:
The smallest value of \(g(x)\) is \(3.5\), which is not listed, so option 4 is correct. \[ \boxed{\text{Option 4: None of the above}} \]
The function \(g(x) = \max(5 - x,\ x + 2)\) takes, at every \(x\), the larger of the two values \(5 - x\) and \(x + 2\). We need the smallest value this larger quantity can ever reach.
The line \(5 - x\) goes down as \(x\) grows, and the line \(x + 2\) goes up as \(x\) grows. So one line is high where the other is low, and the maximum of the two will look like the letter V.
Step 2: Find where the two lines cross:
The V has its lowest point exactly where the two lines meet, so we solve \(5 - x = x + 2\).
\[ 5 - 2 = x + x \Rightarrow 3 = 2x \Rightarrow x = 1.5 \]
At this point both expressions are equal: \(5 - 1.5 = 3.5\) and \(1.5 + 2 = 3.5\).
Step 3: Write g(x) as two pieces:
For \(x \leq 1.5\) we have \(5 - x \geq x + 2\), so \(g(x) = 5 - x\). This piece keeps falling as \(x\) rises, and its smallest value on this side is at \(x = 1.5\), namely \(3.5\).
For \(x \geq 1.5\) we have \(x + 2 \geq 5 - x\), so \(g(x) = x + 2\). This piece keeps rising as \(x\) rises, so its smallest value on this side is also at \(x = 1.5\), namely \(3.5\).
Hence the smallest possible value of \(g(x)\) is \(3.5\).
Step 4: Check option 1:
The value \(4.0\) can be reached by \(g\), but it is not the smallest, because \(g\) comes down to \(3.5\). So option 1 is wrong.
Step 5: Check option 2:
The value \(4.5\) is also reached (for example at \(x = 0.5\), where \(g = 4.5\)), but again it is larger than \(3.5\). So option 2 is wrong.
Step 6: Check option 3:
The number \(1.5\) is the place where the minimum occurs, not the minimum value. Also \(g(x)\) is never below \(3.5\), so it can never be \(1.5\). This is a trap option. So option 3 is wrong.
Step 7: Check option 4:
The true minimum \(3.5\) is not among the first three options, so "None of the above" is correct.
Final Answer:
The smallest value of \(g(x)\) is \(3.5\), which is not listed, so option 4 is correct. \[ \boxed{\text{Option 4: None of the above}} \]
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