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Let the function f be defined by f(x) = 2 - |x - 4| for all real values of x. What is the greatest possible value of f?
- a)-2
- b)2
- c)4
- d)6
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Let the function f be defined by f(x) = 2 - |x - 4| for all real value...
The function f (x) = 2 - |x - 4| reaches its greatest value when the absolute value is minimized. Since absolute values cannot be negative, the least value |x - 4| can have is 0, which it has when x = 4:
f (4) = 2 - |4 - 4| = 2 - 0 = 2
f (4) = 2 - |4 - 4| = 2 - 0 = 2
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Let the function f be defined by f(x) = 2 - |x - 4| for all real value...
Given:
The function f is defined by f(x) = 2 - |x - 4| for all real values of x.
To find:
The greatest possible value of f.
Solution:
To find the greatest possible value of f, we need to maximize the expression 2 - |x - 4|.
Step 1: Break down the expression
The expression 2 - |x - 4| can be broken down into two cases based on the value of (x - 4):
Case 1: If (x - 4) ≥ 0, then |x - 4| = (x - 4).
Case 2: If (x - 4) < 0,="" then="" |x="" -="" 4|="-(x" -="" 4)="4" -="" />
Step 2: Rewrite the expression
Using the two cases, we can rewrite the expression 2 - |x - 4| as:
Case 1: If (x - 4) ≥ 0, then 2 - |x - 4| = 2 - (x - 4) = 2 - x + 4 = 6 - x.
Case 2: If (x - 4) < 0,="" then="" 2="" -="" |x="" -="" 4|="2" -="" (4="" -="" x)="2" -="" 4="" +="" x="x" -="" />
Step 3: Determine the maximum value
For Case 1: 6 - x, the maximum value occurs when x is minimized.
The minimum value of x is 4, so the maximum value of 6 - x is 6 - 4 = 2.
For Case 2: x - 2, the maximum value occurs when x is maximized.
There is no upper limit on x, so the maximum value of x - 2 is infinity (∞).
Step 4: Compare the maximum values
Comparing the maximum values from the two cases, we find that 2 is the greatest possible value of f.
Therefore, the correct answer is option B) 2.
The function f is defined by f(x) = 2 - |x - 4| for all real values of x.
To find:
The greatest possible value of f.
Solution:
To find the greatest possible value of f, we need to maximize the expression 2 - |x - 4|.
Step 1: Break down the expression
The expression 2 - |x - 4| can be broken down into two cases based on the value of (x - 4):
Case 1: If (x - 4) ≥ 0, then |x - 4| = (x - 4).
Case 2: If (x - 4) < 0,="" then="" |x="" -="" 4|="-(x" -="" 4)="4" -="" />
Step 2: Rewrite the expression
Using the two cases, we can rewrite the expression 2 - |x - 4| as:
Case 1: If (x - 4) ≥ 0, then 2 - |x - 4| = 2 - (x - 4) = 2 - x + 4 = 6 - x.
Case 2: If (x - 4) < 0,="" then="" 2="" -="" |x="" -="" 4|="2" -="" (4="" -="" x)="2" -="" 4="" +="" x="x" -="" />
Step 3: Determine the maximum value
For Case 1: 6 - x, the maximum value occurs when x is minimized.
The minimum value of x is 4, so the maximum value of 6 - x is 6 - 4 = 2.
For Case 2: x - 2, the maximum value occurs when x is maximized.
There is no upper limit on x, so the maximum value of x - 2 is infinity (∞).
Step 4: Compare the maximum values
Comparing the maximum values from the two cases, we find that 2 is the greatest possible value of f.
Therefore, the correct answer is option B) 2.
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Let the function f be defined by f(x) = 2 - |x - 4| for all real values of x. What is the greatest possible value of f?a)-2b)2c)4d)6Correct answer is option 'B'. Can you explain this answer?
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