1. **Problem statement:** Approximate the number 1.8 to a natural number $x$ in the interval $\mathbb{R}[-3,3]$ and analyze the function $$f(x) = \frac{5x + 10}{x + 3}$$ 2. **Formula and rules:** The function is a rational function defined for all $x \neq -3$ because the denominator cannot be zero. 3. **Domain:** The domain is $\mathbb{R} \setminus \{-3\}$, but the problem restricts $x$ to the interval $[-3,3]$ excluding $-3$. 4. **Approximate 1.8 to a natural number:** The natural numbers in $[-3,3]$ are $\{0,1,2,3\}$. The closest natural number to 1.8 is $2$. 5. **Evaluate $f(x)$ for $x=2$:** $$f(2) = \frac{5(2) + 10}{2 + 3} = \frac{10 + 10}{5} = \frac{20}{5} = 4$$ 6. **Check if $f(x)$ is defined for all natural $x$ in $[-3,3]$:** - For $x=0$: $$f(0) = \frac{5(0) + 10}{0 + 3} = \frac{10}{3}$$ defined. - For $x=1$: $$f(1) = \frac{5(1) + 10}{1 + 3} = \frac{15}{4}$$ defined. - For $x=2$: as above, defined. - For $x=3$: $$f(3) = \frac{5(3) + 10}{3 + 3} = \frac{25}{6}$$ defined. 7. **Calculate $f(4), f(-4), f(0)$:** - $f(4) = \frac{5(4) + 10}{4 + 3} = \frac{20 + 10}{7} = \frac{30}{7}$ - $f(-4) = \frac{5(-4) + 10}{-4 + 3} = \frac{-20 + 10}{-1} = \frac{-10}{-1} = 10$ - $f(0) = \frac{10}{3}$ as above. 8. **Find intercept with y-axis:** Set $x=0$: $$f(0) = \frac{10}{3}$$ so the y-intercept is $(0, \frac{10}{3})$. 9. **Study variation of $f$ on $[-3,3]$:** - The function has a vertical asymptote at $x=-3$. - For $x > -3$, $f(x)$ is continuous. - Derivative: $$f'(x) = \frac{(5)(x+3) - (5x+10)(1)}{(x+3)^2} = \frac{5x + 15 - 5x - 10}{(x+3)^2} = \frac{5}{(x+3)^2} > 0$$ for all $x \neq -3$. - So $f$ is strictly increasing on $(-3,3]$. 10. **Table of values:** | $x$ | $f(x)$ | |-----|--------| | -2 | $\frac{5(-2)+10}{-2+3} = \frac{-10+10}{1} = 0$ | | 0 | $\frac{10}{3}$ | | 1 | $\frac{15}{4}$ | | 2 | $4$ | | 3 | $\frac{25}{6}$ | 11. **Graphing:** The function has a vertical asymptote at $x=-3$ and is increasing on $(-3,3]$. Plot points from the table and asymptote. 12. **Line segment between $f(3)$ and $f(4)$:** - $f(3) = \frac{25}{6}$ - $f(4) = \frac{30}{7}$ - The segment connects points $(3, \frac{25}{6})$ and $(4, \frac{30}{7})$. **Final answer:** The natural number closest to 1.8 in $[-3,3]$ is $2$, and $f(2) = 4$.