For the following 4 functions, decide if they are linear, exponential,or neither. If a fn is linear or exponential, write down the equation.13f (x) || 500g(x) || 500h(x) || 500u(x)10002000 | 40001000 20001500 20001500 | 2500300025005003500

given data are x 0 1 2 3 f(x) 500 1000 2000 4000 g(x) 500 1000 2000 3000 h(x) 500 1500 2000…

Answered: For the following 4 functions, decide…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For the following 4 functions, decide if they are linear, exponential,
or neither. If a fn is linear or exponential, write down the equation.
1
3
f (x) || 500
g(x) || 500
h(x) || 500
u(x)
1000
2000 | 4000
1000 2000
1500 2000
1500 | 2500
3000
2500
500
3500

Expert Solution

Step 1

given data are

x	0	1	2	3
f(x)	500	1000	2000	4000
g(x)	500	1000	2000	3000
h(x)	500	1500	2000	2500
u(x)	500	1500	2500	3500

for f(x)

$\frac{1000 - 500}{1 - 0} = 500, \frac{2000 - 1000}{2 - 1} = 1000, \frac{4000 - 2000}{3 - 2} = 2000$

slope is not constant so it is not linear. now let $f (x) = a b^{x} + c$ so

$\begin{array}{rcl} 500 & = & a b^{0} + c \\ 500 & = & a + c \\ 1000 & = & a b + c \\ 2000 & = & a b^{2} + c \\ 4000 & = & a b^{3} + c \\ 4000 - 2000 & = & a b^{3} - a b^{2} = a b^{2} (b - 1) \\ 2000 & = & a b^{2} (b - 1) \\ 1000 - 500 & = & a b - a = a (b - 1) \\ 500 & = & a (b - 1) \\ \frac{2000}{500} & = & \frac{a b^{2} (b - 1)}{a (b - 1)} = b^{2} \\ b^{2} & = & 4 \\ b & = & 2, a l l t e r m s a r e p o s i t i v e s o b > 0 \\ a & = & 500 \\ c & = & 0 \\ f (x) & = & 500 \times 2^{x} \end{array}$

f(x) is exponential.

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