Given the functions f(x)=|x-A|-1 and f(1)=-1, find A...

Given the functions $f(x)=|x-A|-1$ and $f(1)=-1,$ find $A$ Solution: since $f(1)=-1,$ the point (-1,1) must satisfy the function. $-1=|-1-A|-1$ Add 1 to both sides of the equation. $\quad|-1-A|=0$ The absolute value of zero is zero, so there is no need for the absolute value signs: $-1-A=0 \Rightarrow A=-1$ This is incorrect. What mistake was made?

Given the functions $f(x)=|x-A|-1$ and $f(1)=-1,$ find $A$
Solution:
since $f(1)=-1,$ the point (-1,1) must satisfy the function. $-1=|-1-A|-1$
Add 1 to both sides of the equation. $\quad|-1-A|=0$
The absolute value of zero is zero, so there is no need for the absolute value signs: $-1-A=0 \Rightarrow A=-1$
This is incorrect. What mistake was made?

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Key Concepts

Function Evaluation

Evaluating a function involves substituting the input value into the function's expression to obtain the corresponding output. It is crucial to correctly plug in the given x-value to ensure that the function is evaluated properly. In problems involving function definitions, precise substitution is key to avoiding errors in calculations.

Error in Misinterpreting Coordinates

A common mistake in solving function problems is misinterpreting the notation or the order of coordinates. One must carefully note that the input to the function (the x-value) should be used in the function's formula, and the corresponding output value should come from the function's evaluation. Confusing the roles of the input and the output can lead to incorrect substitutions and ultimately an erroneous conclusion.

Absolute Value Properties

The absolute value function returns the distance from zero and is always non?negative. When solving equations involving absolute values, isolating the absolute value expression and considering its definition (i.e., that |X| = 0 if and only if X = 0) is essential. This ensures that the equation is solved correctly, taking into account the possible cases that may arise from the absolute value definition.

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