1. **Problem:** Given the base function $y=\frac{1}{x}$, it is transformed by reflecting along the x-axis, vertically stretching by a factor of 3, horizontally stretching by a factor of 4, translating 7 units right, and 5 units down. Find the equation of the transformed function. 2. **Formula and rules:** The base function is $y=\frac{1}{x}$. - Reflection along the x-axis changes $y$ to $-y$. - Vertical stretch by factor 3 multiplies $y$ by 3. - Horizontal stretch by factor 4 replaces $x$ by $\frac{x}{4}$. - Horizontal translation 7 units right replaces $x$ by $x-7$. - Vertical translation 5 units down subtracts 5 from $y$. 3. **Step-by-step transformation:** - Start with $y=\frac{1}{x}$. - Reflect along x-axis: $y = -\frac{1}{x}$. - Vertically stretch by 3: $y = 3 \times \left(-\frac{1}{x}\right) = -\frac{3}{x}$. - Horizontally stretch by 4: replace $x$ by $\frac{x}{4}$, so $$y = -\frac{3}{\frac{x}{4}} = -\frac{3}{\cancel{\frac{x}{4}}} = -\frac{3 \times 4}{x} = -\frac{12}{x}$$ - Horizontally translate 7 units right: replace $x$ by $x-7$, $$y = -\frac{12}{x-7}$$ - Vertically translate 5 units down: subtract 5, $$y = -\frac{12}{x-7} - 5$$ 4. **Domain:** The function is undefined where the denominator is zero, $$x-7 \neq 0 \implies x \neq 7$$ So, domain is all real numbers except $x=7$. 5. **Range:** Since the function is a rational function with vertical stretch and vertical shift, the range is all real numbers except the horizontal asymptote value. The horizontal asymptote is at $y = -5$. So, range is all real numbers except $y = -5$. **Final answers:** - Transformed function: $$y = -\frac{12}{x-7} - 5$$ - Domain: $\{x \in \mathbb{R} \mid x \neq 7\}$ - Range: $\{y \in \mathbb{R} \mid y \neq -5\}$