1. **Solve for exact solutions of** $\frac{1}{x} = 2x + 3$. Multiply both sides by $x$ (assuming $x \neq 0$): $$1 = 2x^2 + 3x$$ Rewrite as a quadratic equation: $$2x^2 + 3x - 1 = 0$$ 2. **Use quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=2$, $b=3$, and $c=-1$. Calculate discriminant: $$\Delta = 3^2 - 4 \times 2 \times (-1) = 9 + 8 = 17$$ 3. **Find roots:** $$x = \frac{-3 \pm \sqrt{17}}{4}$$ Hence, exact solutions are $$x = \frac{-3 + \sqrt{17}}{4}$$ and $$x = \frac{-3 - \sqrt{17}}{4}$$. 4. **Show $f(x) = x^2 - 4x + 9$ is positive for all $x$:** Complete the square: $$f(x) = x^2 - 4x + 9 = (x^2 -4x + 4) + 5 = (x - 2)^2 + 5$$ Since $(x - 2)^2 \geq 0$ for all $x$, and 5 > 0, $f(x) > 0$ for all real $x$. 5. **Minimum point of $f(x)$:** The vertex of a parabola $y = a(x-h)^2 + k$ is at $(h,k)$. Here, $$h = 2, k = 5$$ Minimum point coordinates: $$(2,5)$$. 6. **Sketch graph of $f(x)$ includes intercepts & line of symmetry:** - **Line of symmetry:** $x = 2$. - **$y$-intercept at $x=0$: $$f(0) = 0^2 - 4(0) + 9 = 9$$ so point $(0,9)$. - **Check for $x$-intercepts:** Solve $x^2 - 4x + 9 = 0$, discriminant is $16 - 36 = -20 < 0$, so no real roots. 7. **Find intersection points of** $$y_1 = x^2 - 4x + 2$$ and $$y_2 = -x^2 - 8x$$ Set equal: $$x^2 - 4x + 2 = -x^2 - 8x$$ $$x^2 + x^2 - 4x + 8x + 2 = 0$$ $$2x^2 + 4x + 2 = 0$$ Divide all terms by 2: $$x^2 + 2x + 1 = 0$$ $$(x + 1)^2 = 0$$ So single solution: $x=-1$. Find $y$ coordinate: $$y = (-1)^2 -4(-1) + 2 = 1 + 4 + 2 = 7$$ Intersection point: $(-1, 7)$. 8. **Sketch graph of** $$f(x) = e^{2x} + 2$$ - This is an exponential curve shifted up by 2. - **Horizontal asymptote:** $y = 2$. - The graph increases rapidly as $x$ grows. 9. **Given** $y = \log_2 x$, (a) Express $$\log_2 x^2$$ in terms of $y$: Using log power rule: $$\log_2 x^2 = 2 \log_2 x = 2y$$. (b) The asymptote of $y= \log_2 x$ is the vertical line where the function is undefined: $$x = 0$$. **Final answers:** a) $$x = \frac{-3 \pm \sqrt{17}}{4}$$ b)i) $f(x)$ is positive $orall x\in \mathbb{R}$. ii) Min point: $(2,5)$. iii) Graph described above. c) Intersection point: $(-1,7)$. 2) Sketch described, asymptote $y=2$. 3) a) $\log_2 x^2 = 2y$; b) asymptote: $x=0$.