Domain Range 841666

1. **State the problem:** Find the domain and range of the function $$f(x) = \frac{4x+1}{x-7}$$ in set notation. 2. **Find the domain:** The domain is all real numbers $$x$$ for which the function is defined. - The denominator $$x-7$$ cannot be zero because division by zero is undefined. - So, set $$x-7 \neq 0$$ which gives $$x \neq 7$$. - Therefore, the domain is all real numbers except $$7$$. - In set notation, the domain is $$\{x \in \mathbb{R} \mid x \neq 7\}$$. 3. **Find the range:** The range is all possible values of $$f(x)$$. - Let $$y = \frac{4x+1}{x-7}$$. - Solve for $$x$$ in terms of $$y$$: $$y = \frac{4x+1}{x-7}$$ Multiply both sides by $$x-7$$: $$y(x-7) = 4x + 1$$ $$yx - 7y = 4x + 1$$ Rearrange terms: $$yx - 4x = 7y + 1$$ Factor $$x$$: $$x(y - 4) = 7y + 1$$ $$x = \frac{7y + 1}{y - 4}$$ - For $$x$$ to be defined, the denominator $$y - 4 \neq 0$$, so $$y \neq 4$$. - Therefore, the range is all real numbers except $$4$$. - In set notation, the range is $$\{y \in \mathbb{R} \mid y \neq 4\}$$. 4. **Final answer:** - Domain: $$\{x \in \mathbb{R} \mid x \neq 7\}$$ - Range: $$\{y \in \mathbb{R} \mid y \neq 4\}$$