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Home Question Bank Math Pre Calculus Archive 2025 January 08

Pre-calculus Questions from Jan 08,2025

Browse the Pre-calculus Q&A Archive for Jan 08,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Lepose that the function \( h \) is defined, for all real numbers, as follows. \( h(x)=\left\{\begin{array}{ll}-\frac{1}{2} x+1 & \text { if } x \leq-1 \\ (x-1)^{2}-3 & \text { if }-1<x<2 \\ 1 & \text { if } x \geq 2\end{array}\right. \) \( h(0)=h(2) \), and \( h(4) \) Find the domain and the range of the following function. \( f(x)=\left\{\begin{array}{ll}x+1, & \text { for } x \leq-4 \\ -2, & \text { for }-4<x<5 \\ \frac{1}{5} x, & \text { for } x \geq 5\end{array}\right. \) The domain of the function is Type your answer in interval notation) Determine l'ensemble de definition de la fonctionf definie de IR versIR \[ f(x)=\frac{x+2}{\sqrt{6 x-5}+2 x-3} \] QUESTION 3 The equation of a hyperbola is given by \( f(x)=\frac{3}{x-7}-4 \). Write down the equation of the new function that is forme \( 3.1 \quad \) Shift two units to the left \( 3.2 \quad \) Shift 3 units up \( 3.3 \quad \) Shift 1 unit right and 2 units down \( 3.4 \quad \) The equation of the new hyperbola has new asyr QUESTION 4 Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2} \). Clearly indicate all intercepts with the axes, turning point 2)(Non-calculatorl) The first two terms of a series are \( 1+\sqrt{2} \) and \( 1+\frac{1}{\sqrt{2}} \). (a) If the series is arithmetic, show that the common difference is \( -\frac{1}{2} \sqrt{2} \). Show also that the sum of the first ten terms is \( \frac{5}{2}(4-5 \sqrt{2}) \). (b) If the series is geometric, show that the sum to infinity exists. Show also that \( S_{\infty}=4+3 \sqrt{2} \). Considérons les fonctions \( f \) et \( g \) définies par : \[ g(x)=\sqrt{x+1} \text { et } f(x)=-\frac{1}{2} x^{2}+x+\frac{7}{2} \] The optimal height h of the letters of a messagu printed on pavement is given by the formula \( h=\frac{0.00252 d^{227}}{e} \). Here dis the dietance of the driver from the letters and e is the height of the driver's eye above the pavement All of the distances are in meters. Find \( h \) for the given values of d and e \[ d=1259 \mathrm{~m} \quad \mathrm{e}=14 \mathrm{~m} \] \[ h \approx \] \( \square \) II (Round is the nearest tenth as needed.) 6.1.2 Determine the values of \( x \) in the interval \( 0^{\circ} \leq x \leq 135^{\circ} \) for which \( f(x) \leq-1 \). 10. Write the following in polar form: (i) \( \left.\left.\left[5\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right] \right\rvert\, 2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\right] \) polar (ii) \( \left[3\left(\cos \left(\frac{\pi}{7}\right)+i \sin \left(\frac{\pi}{7}\right)\right)\right]\left[5\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right)\right] \) (iii) \( \left.\left.\left[2\left(\cos \left(\frac{\pi}{4}\right)+i \sin \left(\frac{\pi}{4}\right)\right)\right] \right\rvert\, 7\left(\cos \left(\frac{\pi}{8}\right)+i \sin \left(\frac{\pi}{8}\right)\right)\right] \) (iv) \( \left[7\left(\cos \left(\frac{\pi}{8}\right)+i \sin \left(\frac{\pi}{8}\right)\right) \left\lvert\,\left[9\left(\cos \left(\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{3}\right)\right)\right]\right.\right. \) (v) \( \left.\left[9\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\right] \left\lvert\, 3\left(\cos \left(\frac{\pi}{5}\right)+i \sin \left(\frac{\pi}{5}\right)\right)\right.\right] \) Prove that \( a+a r+a r^{2}+\ldots( \) to \( n \) terms \( )=\frac{a\left(1-r^{n}\right)}{1-r} \) for \( T \neq 1 \). Given the geometric series \( 15+5+\frac{5}{3}+\ldots \) \( 5.2 .1 \quad \) Explain why the series converges. \( 5.2 .2 \quad \) Evaluate \( \sum_{n=1}^{\infty} 5\left(3^{2-n}\right) \) The sum of the first \( n \) terms of a sequence is given by \( S_{n}=2^{n+2}-4 \). \( 5.3 .1 \quad \) Determine the sum of the first 24 terms.
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