1. **State the problem:** We have two functions $f(x) = 6x + 5$ and $g(x) = 4 - 9x$. 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 $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, $(ff)(x)$, $\left(\frac{f}{g}\right)(x)$, and $\left(\frac{g}{f}\right)(x)$. 2. **Domain rules:** The domain of a function is all $x$ values for which the function is defined. - For polynomials and linear functions like $f$ and $g$, the domain is all real numbers $(-\infty, \infty)$. - For sums, differences, and products of functions, the domain is the intersection of the domains of the individual functions. - For quotients $\frac{f}{g}$ or $\frac{g}{f}$, the domain excludes values where the denominator is zero. 3. **Find domains:** - $f(x) = 6x + 5$ is linear, so domain is $(-\infty, \infty)$. - $g(x) = 4 - 9x$ is linear, so domain is $(-\infty, \infty)$. - $f+g$, $f-g$, $fg$, and $ff$ are combinations of $f$ and $g$, so their domains are $(-\infty, \infty)$. - For $\frac{f}{g}$, domain excludes $x$ where $g(x) = 0$: $$4 - 9x = 0 \implies 9x = 4 \implies x = \frac{4}{9}$$ So domain is $(-\infty, \infty) \setminus \left\{ \frac{4}{9} \right\}$. - For $\frac{g}{f}$, domain excludes $x$ where $f(x) = 0$: $$6x + 5 = 0 \implies 6x = -5 \implies x = -\frac{5}{6}$$ So domain is $(-\infty, \infty) \setminus \left\{ -\frac{5}{6} \right\}$. 4. **Find expressions:** - $(f+g)(x) = f(x) + g(x) = (6x + 5) + (4 - 9x) = 6x + 5 + 4 - 9x = (6x - 9x) + (5 + 4) = -3x + 9$ - $(f-g)(x) = f(x) - g(x) = (6x + 5) - (4 - 9x) = 6x + 5 - 4 + 9x = (6x + 9x) + (5 - 4) = 15x + 1$ - $(fg)(x) = f(x) \cdot g(x) = (6x + 5)(4 - 9x) = 6x \cdot 4 + 6x \cdot (-9x) + 5 \cdot 4 + 5 \cdot (-9x) = 24x - 54x^2 + 20 - 45x = -54x^2 + (24x - 45x) + 20 = -54x^2 - 21x + 20$ - $(ff)(x) = f(f(x)) = f(6x + 5) = 6(6x + 5) + 5 = 36x + 30 + 5 = 36x + 35$ - $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{6x + 5}{4 - 9x}$ with domain excluding $x = \frac{4}{9}$. - $\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{4 - 9x}{6x + 5}$ with domain excluding $x = -\frac{5}{6}$. 5. **Summary:** - Domains: - $f, g, f+g, f-g, fg, ff$: $(-\infty, \infty)$ - $\frac{f}{g}$: $(-\infty, \infty) \setminus \left\{ \frac{4}{9} \right\}$ - $\frac{g}{f}$: $(-\infty, \infty) \setminus \left\{ -\frac{5}{6} \right\}$ - Expressions: - $(f+g)(x) = -3x + 9$ - $(f-g)(x) = 15x + 1$ - $(fg)(x) = -54x^2 - 21x + 20$ - $(ff)(x) = 36x + 35$ - $\left(\frac{f}{g}\right)(x) = \frac{6x + 5}{4 - 9x}$ - $\left(\frac{g}{f}\right)(x) = \frac{4 - 9x}{6x + 5}$