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Home Question Bank Math Pre Calculus Archive 2025 January 18

Pre-calculus Questions from Jan 18,2025

Browse the Pre-calculus Q&A Archive for Jan 18,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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8. \( f(x)=\sqrt{\frac{x^{2}-1}{x^{2}+1}} \) y \( g(x)=\sqrt{\frac{x+1}{x-1}} \) 9. \( f(x)=\{(2,5),(3,6),(4,7),(5,8)\} \quad \) y \( \quad g(x)=\{(1,2),(2,3),(3,4),(4,5)\} \) (b) Determine the value of \( m \) in each of the following: (1) \( \sum_{k=1}^{m}(k+1)=1595 \) (2) \( \sum_{r=1}^{m} 5^{r}=19530 \) (3) \( \sum_{p=3}^{m}(4 p-5)=7500 \) (4) \( \sum_{p=5}^{m}(2-3 p)=-7085 \) (5) \( \sum_{i=4}^{m} 8\left(\frac{2}{3}\right)^{i-5}=\frac{2660}{81} \) (6) \( \sum_{k=-2}^{m} \frac{1}{2}(3)^{k-1}=\frac{1640}{27} \) (7) \( \sum_{k=1}^{m}(6 k-8)<2600 \) (8) \( \sum_{r=2}^{m} 2(3-r)>-800 \) (9) \( \sum_{p=0}^{m} 3(2)^{2 p-3}<8000 \) (c) Write the following series in sigma notation: (1) \( -3+4+11+18+\ldots+137 \) (2) \( 2+2+2+2+\ldots+2(15 \) terms \( ) \) (3) \( 6+2+\frac{2}{3}+\frac{2}{9}+\ldots \) (to infinity) (4) \( 10+6+2-2-\ldots-210 \) (5) \( -2+14-98+686-\ldots+33614 \) (6) \( 3+3+5+9+\ldots+555 \) (7) \( x+(x+3)+(x+6)+(x+9)+\ldots+(x+120) \) (8) \( 4+4 x+4 x^{2}+4 x^{3}+\ldots+4 x^{65} \) (d) For which values of \( x \) will the geometric series \( \sum_{i=1}^{\infty} 4(3-x)^{i} \) converge? (e) Calculate the value of \( p \) if \( \sum_{k=0}^{\infty} 27 p^{k+1}=\sum_{r=3}^{14}(30-3 r) \). Question 5: \( (5 \) Marks \( ) \) Obtain the inverse \( z \)-transform by finding \( x(k) \) for \( k=0,1,2,3,4 \) of the function: \( x(z)=\frac{10 z+5}{(z-1)(z-0.2)} \) Explain how the graph of the function \( h(x) = -\sqrt{x} \) is affected by the transformation compared to the original \( f(x) = \sqrt{x} \). Sketch gmex of \( f(x)=\frac{3 x^{2}+2}{x^{2}+4 x} \) sketch graph of \( f(x)=\frac{3 x^{2}+2}{x^{2}+4 x} \) \( f(x)=\sqrt {x+\sqrt {x+\sqrt {x}}} \) EASTERN CAPE MAY/SUNE 12023 Given the following geometric series: \( \frac{24}{x}+12+6 x+3 x^{2}+\ldots \) 3.1.1 Determine the value of \( r \), the common ratio, in terms of \( x \) 3.1.2 Determine the value of \( x \) for which this series converges 3.1.3 If \( x=4 \), determine the sum of the series to 15 terms. Calculate: \( \sum_{n=1} 6(2)^{-4} \) The sum of the first \( n \) terms of an arithmetic series is given \( S_{n}= \) 3.3.1 Calculate the sum of the first 15 terms 6. Sketch the graph of: a) \( f(x)=\frac{x^{2}+x-6}{x^{3}-4 x^{2}-7 x+10} \quad \) b) \( f(x)=\frac{3 x^{2}-2 x+1}{x-1} \) c) \( f(x)=\frac{3 x^{2}+2}{x^{2}+4 x-5} \) What is the range of the inverse function for the function \( f(x) = \sqrt{x - 4} + 1 \)?
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