Function Domains 7B5Daa

1. **State the problem:** We have two functions $f(x) = x - 8$ and $g(x) = \sqrt{x + 9}$. We need to find the domains of $f$, $g$, $f+g$, $f-g$, $fg$, $ff$, $\frac{f}{g}$, and $\frac{g}{f}$, and then find the expressions for these combined functions. 2. **Find the domain of $f(x)$:** - $f(x) = x - 8$ is a linear function defined for all real numbers. - So, $\text{Domain}(f) = (-\infty, \infty)$. 3. **Find the domain of $g(x)$:** - $g(x) = \sqrt{x + 9}$ requires the radicand to be non-negative: $$x + 9 \geq 0$$ $$x \geq -9$$ - So, $\text{Domain}(g) = [-9, \infty)$. 4. **Find the domain of $f+g$ and $f-g$:** - Both require $x$ to be in the domains of $f$ and $g$. - Intersection of domains: $(-\infty, \infty) \cap [-9, \infty) = [-9, \infty)$. - So, $\text{Domain}(f+g) = \text{Domain}(f-g) = [-9, \infty)$. 5. **Find the domain of $fg$:** - Product requires $x$ in domains of both $f$ and $g$. - So, $\text{Domain}(fg) = [-9, \infty)$. 6. **Find the domain of $ff$ (composition $f(f(x))$):** - $f(f(x)) = f(x - 8) = (x - 8) - 8 = x - 16$. - Since $f$ is defined for all real $x$, $ff$ is defined for all real $x$. - So, $\text{Domain}(ff) = (-\infty, \infty)$. 7. **Find the domain of $\frac{f}{g}$:** - Requires $x$ in domains of $f$ and $g$ and $g(x) \neq 0$. - From $g(x) = \sqrt{x + 9}$, $g(x) = 0$ when $x + 9 = 0 \Rightarrow x = -9$. - So domain is $[-9, \infty)$ excluding $x = -9$. - Hence, $\text{Domain}\left(\frac{f}{g}\right) = (-9, \infty)$. 8. **Find the domain of $\frac{g}{f}$:** - Requires $x$ in domains of $g$ and $f$ and $f(x) \neq 0$. - $f(x) = x - 8 = 0$ when $x = 8$. - So domain is $[-9, \infty)$ excluding $x = 8$. - Hence, $\text{Domain}\left(\frac{g}{f}\right) = [-9, 8) \cup (8, \infty)$. 9. **Find $(f+g)(x)$:** $$ (f+g)(x) = f(x) + g(x) = (x - 8) + \sqrt{x + 9} $$ 10. **Find $(f-g)(x)$:** $$ (f-g)(x) = f(x) - g(x) = (x - 8) - \sqrt{x + 9} $$ 11. **Find $(fg)(x)$:** $$ (fg)(x) = f(x) \cdot g(x) = (x - 8) \sqrt{x + 9} $$ 12. **Find $(ff)(x)$:** $$ (ff)(x) = f(f(x)) = f(x - 8) = (x - 8) - 8 = x - 16 $$ 13. **Find $\left(\frac{f}{g}\right)(x)$:** $$ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x - 8}{\sqrt{x + 9}} $$ 14. **Find $\left(\frac{g}{f}\right)(x)$:** $$ \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{\sqrt{x + 9}}{x - 8} $$ **Final answers:** - $\text{Domain}(f) = (-\infty, \infty)$ - $\text{Domain}(g) = [-9, \infty)$ - $\text{Domain}(f+g) = \text{Domain}(f-g) = \text{Domain}(fg) = [-9, \infty)$ - $\text{Domain}(ff) = (-\infty, \infty)$ - $\text{Domain}\left(\frac{f}{g}\right) = (-9, \infty)$ - $\text{Domain}\left(\frac{g}{f}\right) = [-9, 8) \cup (8, \infty)$ - $(f+g)(x) = x - 8 + \sqrt{x + 9}$ - $(f-g)(x) = x - 8 - \sqrt{x + 9}$ - $(fg)(x) = (x - 8) \sqrt{x + 9}$ - $(ff)(x) = x - 16$ - $\left(\frac{f}{g}\right)(x) = \frac{x - 8}{\sqrt{x + 9}}$ - $\left(\frac{g}{f}\right)(x) = \frac{\sqrt{x + 9}}{x - 8}$