1. **Problem Statement:** Find the natural domain and analyze the functions given in exercises 15 to 20. 2. **Function 15: $f(x) = 5 - 2x$** - This is a linear function. - The natural domain of any linear function is all real numbers, so $\text{Domain} = (-\infty, \infty)$. 3. **Function 16: $f(x) = 1 - 2x - x^2$** - This is a quadratic function. - Quadratic functions are defined for all real numbers. - So, $\text{Domain} = (-\infty, \infty)$. 4. **Function 17: $g(x) = \sqrt{|x|}$** - The expression inside the square root must be $\geq 0$. - Since $|x| \geq 0$ for all $x$, the domain is all real numbers. - So, $\text{Domain} = (-\infty, \infty)$. 5. **Function 18: $g(x) = \sqrt{-x}$** - The expression inside the square root must be $\geq 0$. - So, $-x \geq 0 \Rightarrow x \leq 0$. - Domain is all real numbers less than or equal to zero. - So, $\text{Domain} = (-\infty, 0]$. 6. **Function 19: $F(t) = \frac{t}{|t|}$** - The denominator $|t|$ cannot be zero. - So, $t \neq 0$. - Domain is all real numbers except zero. - So, $\text{Domain} = (-\infty, 0) \cup (0, \infty)$. 7. **Function 20: $G(t) = \frac{1}{|t|}$** - The denominator $|t|$ cannot be zero. - So, $t \neq 0$. - Domain is all real numbers except zero. - So, $\text{Domain} = (-\infty, 0) \cup (0, \infty)$. **Summary:** - Functions 15 and 16 have domain $(-\infty, \infty)$. - Function 17 has domain $(-\infty, \infty)$. - Function 18 has domain $(-\infty, 0]$. - Functions 19 and 20 have domain $(-\infty, 0) \cup (0, \infty)$. **Graph notes:** - Linear and quadratic functions are smooth curves. - Square root functions start at the boundary of their domain and extend rightwards. - Functions with absolute values in denominators have vertical asymptotes at zero.