To determine the range of the function [tex]\( f(x) = \frac{-4}{x} + 1 \)[/tex], we follow these steps:
1. Express the function:
[tex]\[ y = \frac{-4}{x} + 1. \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-4}{x} + 1. \][/tex]
Subtract 1 from both sides:
[tex]\[ y - 1 = \frac{-4}{x}. \][/tex]
To isolate [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[ x(y - 1) = -4. \][/tex]
[tex]\[ x = \frac{-4}{y - 1}. \][/tex]
3. Determine when the function is undefined:
For [tex]\( x \)[/tex] to be defined, the denominator [tex]\( y - 1 \)[/tex] in [tex]\( x = \frac{-4}{y - 1} \)[/tex] must not be zero:
[tex]\[ y - 1 \neq 0. \][/tex]
[tex]\[ y \neq 1. \][/tex]
Therefore, the function [tex]\( f(x) \)[/tex] is undefined when [tex]\( y = 1 \)[/tex].
4. Identify the range:
The function's range is all real numbers except where the function is undefined. Hence, the function [tex]\( f(x) \)[/tex] can take any real value except [tex]\( 1 \)[/tex].
Thus, the range of the function [tex]\( f(x) = \frac{-4}{x} + 1 \)[/tex] is:
[tex]\[ (-\infty, 1) \cup (1, \infty). \][/tex]
Therefore, the correct answer is B. [tex]\( (-\infty, 1) \cup (1, \infty) \)[/tex].
