To determine how the range of [tex]\( g(x) = \frac{6}{x} \)[/tex] compares with the range of the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], we should analyze the mathematical properties of these functions.
1. Parent Function [tex]\( f(x) = \frac{1}{x} \)[/tex]:
- This is a classic rational function.
- The domain of [tex]\( f(x) \)[/tex] is all real numbers except [tex]\( x = 0 \)[/tex] because division by zero is undefined.
- To find the range, we consider the output values [tex]\( y \)[/tex] for different inputs [tex]\( x \)[/tex].
- For any nonzero [tex]\( x \)[/tex], [tex]\( f(x) = \frac{1}{x} \)[/tex] will also yield a nonzero output. It captures all possible values except zero. Therefore, the range of [tex]\( f(x) \)[/tex] is all nonzero real numbers.
2. Function [tex]\( g(x) = \frac{6}{x} \)[/tex]:
- This function is similar to [tex]\( f(x) \)[/tex] but with a different constant numerator.
- The domain of [tex]\( g(x) \)[/tex] is likewise all real numbers except [tex]\( x = 0 \)[/tex] since division by zero is undefined.
- To find the range, let's consider the output values [tex]\( y \)[/tex] when we input different [tex]\( x \)[/tex].
- For any nonzero [tex]\( x \)[/tex], [tex]\( g(x) = \frac{6}{x} \)[/tex] will also produce a nonzero output. Just like [tex]\( f(x) \)[/tex], it can take any nonzero value [tex]\( y \)[/tex]. Thus, the range of [tex]\( g(x) \)[/tex] is all nonzero real numbers.
3. Comparison of Ranges:
- The range of [tex]\( f(x) = \frac{1}{x} \)[/tex] is all nonzero real numbers.
- The range of [tex]\( g(x) = \frac{6}{x} \)[/tex] is also all nonzero real numbers.
Therefore, the correct comparison is:
"The range of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is all nonzero real numbers."
Hence, the correct option is:
- The range of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is all nonzero real numbers.
