how does the range of $g(x) = \frac{6}{x}$ compare with the range of the parent function $f(x) = \frac{1}{x}$?

- the range of both $f(x)$ and $g(x)$ is all real numbers
- the range of both $f(x)$ and $g(x)$ is all nonzero real numbers
- the range of $f(x)$ is all real numbers, the range of $g(x)$ is all real numbers except 6
- the range of $f(x)$ is all nonzero real numbers, the range of $g(x)$ is all real numbers except 6

Question

how does the range of $g(x) = \frac{6}{x}$ compare with the range of the parent function $f(x) = \frac{1}{x}$?

- the range of both $f(x)$ and $g(x)$ is all real numbers
- the range of both $f(x)$ and $g(x)$ is all nonzero real numbers
- the range of $f(x)$ is all real numbers, the range of $g(x)$ is all real numbers except 6
- the range of $f(x)$ is all nonzero real numbers, the range of $g(x)$ is all real numbers except 6

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Range of Rational Functions", "Rational Function Transformations", "Parent Function Comparison" ], "new_concepts": [], "current_concepts": [ "Range of Rational Functions", "Rational Function Transformations", "Parent Function Comparison" ] } ## Determine range of parent function Using the *Range of Rational Functions* knowledge point $$ \begin{aligned} &f(x) = \frac{1}{x} \ &y = \frac{1}{x} \implies x = \frac{1}{y} \implies y eq 0 \ &\text{Range of } f(x): \{y \in \mathbb{R} \mid y eq 0\} \end{aligned} $$ ## Determine range of transformed function Using the *Rational Function Transformations* and *Range of Rational Functions* knowledge points $$ \begin{aligned} &g(x) = \frac{6}{x} = 6 \cdot f(x) \ &y = \frac{6}{x} \implies x = \frac{6}{y} \implies y eq 0 \ &\text{Range of } g(x): \{y \in \mathbb{R} \mid y eq 0\} \end{aligned} $$ ## Compare the ranges Using the *Parent Function Comparison* knowledge point $$ \begin{aligned} &\text{Range of } f(x) = \text{all nonzero real numbers} \ &\text{Range of } g(x) = \text{all nonzero real numbers} \end{aligned} $$

The range of both f(x) and g(x) is all real numbers

The range of both f(x) and g(x) is all nonzero real numbers

The range of f(x) is all real numbers, the range of g(x) is all real numbers except 6

The range of f(x) is all nonzero real numbers, the range of g(x) is all real numbers except 6

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Range of Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Determine range of parent function

Determine range of transformed function

Compare the ranges

Answer: