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20 May 2019, 00:13
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The function f is defined for all numbers x by f(x) = |x + 3| + |x + 2|. For which value of x, does f(x) = f(x - 1) ?
A. -3
B. -2
C. -1
D. 1
E. 2
M37-69
A. -3
B. -2
C. -1
D. 1
E. 2
M37-69
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30 Nov 2021, 06:29
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Official Solution:
The function \(f\) is defined for all numbers \(x\) by \(f(x) = |x + 3| + |x + 2|\). For which value of \(x\), does \(f(x) = f(x - 1)\) ?
A. \(-3\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)
\(f(x) = |x + 3| + |x + 2|\).
\(f(x-1) = |(x-1) + 3| + |(x-1) + 2|= |x + 2| + |x + 1|\).
We should find such \(x\) that \(|x + 3| + |x + 2|=|x + 2| + |x + 1|\):
Cancel \(|x + 2|\) to get \(|x + 3| = |x + 1|\).
The above mean that, on the number line, \(x\) is equidistant from -3 and -1, so \(x=\frac{-3 + (-1)}{2}=-2\).
Answer: B
The function \(f\) is defined for all numbers \(x\) by \(f(x) = |x + 3| + |x + 2|\). For which value of \(x\), does \(f(x) = f(x - 1)\) ?
A. \(-3\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)
\(f(x) = |x + 3| + |x + 2|\).
\(f(x-1) = |(x-1) + 3| + |(x-1) + 2|= |x + 2| + |x + 1|\).
We should find such \(x\) that \(|x + 3| + |x + 2|=|x + 2| + |x + 1|\):
Cancel \(|x + 2|\) to get \(|x + 3| = |x + 1|\).
The above mean that, on the number line, \(x\) is equidistant from -3 and -1, so \(x=\frac{-3 + (-1)}{2}=-2\).
Answer: B
General Discussion
20 May 2019, 01:20
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Given
The function f(x) = |x + 3| + |x + 2| ...... Equation A
To Find
For which value of x, does f(x) = f(x - 1) ?
Let us replace "x" by "x-1" in the function f(x) to get the function f(x-1):
f(x-1) = | (x-1) + 3 | + | (x-1) + 2 |
f(x-1) = |x+2| + |x+1| ...... Equation B
To find the value of x:
Equation A = Equation B
|x + 3| + |x + 2| = |x+2| + |x+1|
|x+3| = |x+1| ...... Equation C
Now, let us start substituting the values from the options to check if it fits the Equation c
1) When x = -3
|-3+3| = |-3+1|
|0| = |-2|
0 = 2 (Absurd)
So x cannot be -3
2) When x = -2
|-2+3| = |-2+1|
|1| = |-1|
1 = 1 ( Makes sense )
Since LHS = RHS , x=-2 is a sloution of the Equation C.
We do not need to check for the rest of the options as we have already got our answer, x=-2
So the correct answer is B.
The function f(x) = |x + 3| + |x + 2| ...... Equation A
To Find
For which value of x, does f(x) = f(x - 1) ?
Let us replace "x" by "x-1" in the function f(x) to get the function f(x-1):
f(x-1) = | (x-1) + 3 | + | (x-1) + 2 |
f(x-1) = |x+2| + |x+1| ...... Equation B
To find the value of x:
Equation A = Equation B
|x + 3| + |x + 2| = |x+2| + |x+1|
|x+3| = |x+1| ...... Equation C
Now, let us start substituting the values from the options to check if it fits the Equation c
1) When x = -3
|-3+3| = |-3+1|
|0| = |-2|
0 = 2 (Absurd)
So x cannot be -3
2) When x = -2
|-2+3| = |-2+1|
|1| = |-1|
1 = 1 ( Makes sense )
Since LHS = RHS , x=-2 is a sloution of the Equation C.
We do not need to check for the rest of the options as we have already got our answer, x=-2
So the correct answer is B.
