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Given that $f(x) = 2e^{x} - 5, \; x \in \mathbb{R}$ (a) sketch, on separate diagrams, the curve with equation (i) $y = f(x)$ (ii) $y = |f(x)|$ On each diagram, show the coordinates of each point at which the curve meets or cuts the axes - Edexcel - A-Level Maths Pure - Question 3 - 2015 - Paper 3

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Given-that--$f(x)-=-2e^{x}---5,-\;-x-\in-\mathbb{R}$--(a)-sketch,-on-separate-diagrams,-the-curve-with-equation--(i)-$y-=-f(x)$--(ii)-$y-=-|f(x)|$--On-each-diagram,-show-the-coordinates-of-each-point-at-which-the-curve-meets-or-cuts-the-axes-Edexcel-A-Level Maths Pure-Question 3-2015-Paper 3.png

Given that $f(x) = 2e^{x} - 5, \; x \in \mathbb{R}$ (a) sketch, on separate diagrams, the curve with equation (i) $y = f(x)$ (ii) $y = |f(x)|$ On each diagram, ... show full transcript

Worked Solution & Example Answer:Given that $f(x) = 2e^{x} - 5, \; x \in \mathbb{R}$ (a) sketch, on separate diagrams, the curve with equation (i) $y = f(x)$ (ii) $y = |f(x)|$ On each diagram, show the coordinates of each point at which the curve meets or cuts the axes - Edexcel - A-Level Maths Pure - Question 3 - 2015 - Paper 3

Step 1

Find the exact solutions of the equation |f(x)| = 2

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To solve ∣f(x)∣=2|f(x)| = 2:

  1. Considering f(x)=2f(x) = 2: 2ex−5=2  ⟹  2ex=7  ⟹  ex=72  ⟹  x=ln⁡(72).2e^{x} - 5 = 2 \implies 2e^{x} = 7 \implies e^{x} = \frac{7}{2} \implies x = \ln(\frac{7}{2}).

  2. Considering f(x)=−2f(x) = -2: 2ex−5=−2  ⟹  2ex=3  ⟹  ex=32  ⟹  x=ln⁡(32).2e^{x} - 5 = -2 \implies 2e^{x} = 3 \implies e^{x} = \frac{3}{2} \implies x = \ln(\frac{3}{2}).

Therefore, the exact solutions are: x=ln⁡(72),…,ln⁡(32).x = \ln(\frac{7}{2}), \ldots, \ln(\frac{3}{2}).

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