1. **Problem 15: Find the natural domain and graph of** $f(x) = 5 - 2x$. - This is a linear function. - The natural domain of any linear function is all real numbers, so $\text{Domain} = (-\infty, \infty)$. 2. **Problem 16: Find the natural domain and graph of** $f(x) = 1 - 2x - x^2$. - This is a quadratic function. - Quadratic functions are defined for all real numbers. - So, $\text{Domain} = (-\infty, \infty)$. 3. **Problem 17: Find the natural domain and graph of** $g(x) = \sqrt{|x|}$. - The expression inside the square root must be $\geq 0$. - Since $|x| \geq 0$ for all $x$, the domain is all real numbers. - So, $\text{Domain} = (-\infty, \infty)$. 4. **Problem 18: Find the natural domain and graph of** $g(x) = \sqrt{-x}$. - The expression inside the square root must be $\geq 0$. - So, $-x \geq 0 \Rightarrow x \leq 0$. - The domain is $(-\infty, 0]$. 5. **Problem 19: Find the natural domain and graph of** $F(t) = \frac{t}{|t|}$. - The denominator $|t|$ cannot be zero. - So, $t \neq 0$. - Domain is $(-\infty, 0) \cup (0, \infty)$. - The function equals $-1$ for $t<0$ and $1$ for $t>0$. 6. **Problem 20: Find the natural domain and graph of** $G(t) = \frac{1}{|t|}$. - The denominator $|t|$ cannot be zero. - So, $t \neq 0$. - Domain is $(-\infty, 0) \cup (0, \infty)$. - The graph has two branches, both positive, approaching infinity near $t=0$ and approaching zero as $|t|$ grows. **Summary of domains:** - 15: $(-\infty, \infty)$ - 16: $(-\infty, \infty)$ - 17: $(-\infty, \infty)$ - 18: $(-\infty, 0]$ - 19: $(-\infty, 0) \cup (0, \infty)$ - 20: $(-\infty, 0) \cup (0, \infty)$