1. **Problem Statement:** Given functions $f(x) = x + 2$ and $g(x) = \frac{1}{x-9}$, find and analyze the composite function $h(x) = f(g(x))$. 2. **Find the equation of $h(x)$:** By definition, $h(x) = f(g(x)) = f\left(\frac{1}{x-9}\right)$. Since $f(x) = x + 2$, substitute $g(x)$ into $f$: $$h(x) = \frac{1}{x-9} + 2$$ 3. **Simplify $h(x)$:** Write $2$ as $\frac{2(x-9)}{x-9}$ to combine terms: $$h(x) = \frac{1}{x-9} + \frac{2(x-9)}{x-9} = \frac{1 + 2(x-9)}{x-9}$$ Simplify numerator: $$1 + 2(x-9) = 1 + 2x - 18 = 2x - 17$$ So, $$h(x) = \frac{2x - 17}{x - 9}$$ 4. **Domain of $h(x)$:** The domain excludes values making denominator zero: $$x - 9 \neq 0 \implies x \neq 9$$ So, domain is all real numbers except $9$. 5. **Range of $h(x)$:** Set $y = \frac{2x - 17}{x - 9}$ and solve for $x$: $$y(x - 9) = 2x - 17$$ $$yx - 9y = 2x - 17$$ Group $x$ terms: $$yx - 2x = 9y - 17$$ $$x(y - 2) = 9y - 17$$ $$x = \frac{9y - 17}{y - 2}$$ For $x$ to be defined, denominator $y - 2 \neq 0$, so $y \neq 2$. Thus, range is all real numbers except $2$. 6. **Intercepts:** - **x-intercept:** Set $h(x) = 0$: $$\frac{2x - 17}{x - 9} = 0 \implies 2x - 17 = 0 \implies x = \frac{17}{2} = 8.5$$ - **y-intercept:** Set $x=0$: $$h(0) = \frac{2(0) - 17}{0 - 9} = \frac{-17}{-9} = \frac{17}{9} \approx 1.89$$ 7. **Summary:** $$h(x) = \frac{2x - 17}{x - 9}$$ Domain: $x \neq 9$ Range: $y \neq 2$ x-intercept: $(8.5, 0)$ y-intercept: $(0, \frac{17}{9})$ 8. **Graph features:** - Vertical asymptote at $x=9$ - Horizontal asymptote at $y=2$ - Intercepts as above This rational function has a hyperbola shape with these asymptotes and intercepts.