To determine whether the equation [tex]\( |x| - y = 1 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], let’s follow a step-by-step approach:
1. Rewrite the Equation:
The given equation is:
[tex]\[
|x| - y = 1
\][/tex]
2. Isolate [tex]\( y \)[/tex]:
To see if [tex]\( y \)[/tex] can be written explicitly in terms of [tex]\( x \)[/tex], we isolate [tex]\( y \)[/tex] in the equation:
[tex]\[
|x| - 1 = y \; \Rightarrow \; y = |x| - 1
\][/tex]
3. Analysis:
The equation [tex]\( y = |x| - 1 \)[/tex] expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex]. This means for each value of [tex]\( x \)[/tex], there is exactly one value of [tex]\( y \)[/tex].
To check this further:
- When [tex]\( x \geq 0 \)[/tex], [tex]\( y = x - 1 \)[/tex].
- When [tex]\( x < 0 \)[/tex], [tex]\( y = -x - 1 \)[/tex].
In both cases, you get one unique value of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
4. Function Definition:
A function is defined such that for every input [tex]\( x \)[/tex], there is exactly one output [tex]\( y \)[/tex]. From the analysis, [tex]\( y = |x| - 1 \)[/tex] guarantees that each [tex]\( x \)[/tex] maps to only one [tex]\( y \)[/tex].
5. Conclusion:
Since [tex]\( y = |x| - 1 \)[/tex] provides a unique [tex]\( y \)[/tex] for every [tex]\( x \)[/tex], the given equation [tex]\( |x| - y = 1 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Therefore, the answer to the question is:
[tex]\[
\text{Yes}
\][/tex]
