Alright, let's carefully explore the function [tex]\( f(x) \)[/tex] defined as follows:
[tex]\[ f(x) =
\begin{cases}
\frac{x^2 - x}{x - 1} & \text{if } x \neq 1, \\
2 & \text{if } x = 1.
\end{cases}
\][/tex]
### Step-by-Step Analysis
1. Evaluate [tex]\( f(x) \)[/tex] for [tex]\( x \neq 1 \)[/tex]:
Let's simplify the expression [tex]\(\frac{x^2 - x}{x - 1}\)[/tex].
The numerator [tex]\( x^2 - x \)[/tex] can be factored:
[tex]\[ x^2 - x = x(x - 1). \][/tex]
Substituting this back into the fraction:
[tex]\[ \frac{x^2 - x}{x - 1} = \frac{x(x - 1)}{x - 1}. \][/tex]
For [tex]\( x \neq 1 \)[/tex], the [tex]\((x - 1)\)[/tex] terms in the numerator and denominator cancel out:
[tex]\[ \frac{x(x - 1)}{x - 1} = x. \][/tex]
Therefore, for [tex]\( x \neq 1 \)[/tex]:
[tex]\[ f(x) = x. \][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] specifically at [tex]\( x = 1 \)[/tex]:
By definition in the function:
[tex]\[ f(1) = 2. \][/tex]
3. Verify the consistency and overall understanding:
- For [tex]\( x \neq 1 \)[/tex], [tex]\( f(x) = x \)[/tex].
- At [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 2 \)[/tex].
4. Example Value Testing:
Let's test the function for a value other than 1 to verify consistency (e.g., [tex]\( x = 2 \)[/tex]):
Substitute [tex]\( x = 2 \)[/tex] into the simplified form:
[tex]\[ f(2) = 2. \][/tex]
### Conclusion
Combining all the information:
- If [tex]\( x \neq 1 \)[/tex], then [tex]\( f(x) = x \)[/tex]. For [tex]\( x = 2 \)[/tex], the function value is [tex]\( f(2) = 2 \)[/tex].
- At [tex]\( x = 1 \)[/tex], the function specifies directly that [tex]\( f(1) = 2 \)[/tex].
Thus, the results for the given conditions are:
[tex]\[ f(1) = 2 \][/tex]
[tex]\[ f(2) = 2 \][/tex]
These results help us confirm that the function behaves as expected at and around [tex]\( x = 1 \)[/tex].
