Question

name:
linear or nonlinear functions
sheet 1
a) determine whether each function table is linear or nonlinear.
1)

x	-16	-3	11	15
f(x)	-24	-11	3	7

2)

x	1	4	6	7
f(x)	-7	23	63	89

3)

x	-7	0	9	12
f(x)	-10	-3	6	9

4)

x	-9	-6	-1	14
f(x)	-7	-7	2	16

5)

x	-3	2	13	17
f(x)	-21	-26	59	41

6)

x	-14	-10	-5	-4
f(x)	-11	-7	-2	-1

b) 1) which of the following tables represents a linear function?
a)

x	-2	0	6	9
f(x)	0	-4	45	96

b)

x	3	7	11	14
f(x)	17	37	57	72

c)

x	-7	-6	1	3
f(x)	40	16	0	7

1. which of the following tables represents a nonlinear function?

a)

x	1	3	9	10
f(x)	-3	-1	5	6

b)

x	-10	2	15	20
f(x)	-51	9	74	99

c)

x	-8	-3	4
f(x)	65	2	17

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Explanation:

Step1: Check constant x & Δf(x)/Δx

For a linear function, $\frac{\Delta f(x)}{\Delta x}$ (rate of change) must be constant.
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Problem A1

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-3-(-16)=13$, $\Delta f(x)_1=-11-(-24)=13$; $\Delta x_2=11-(-3)=14$, $\Delta f(x)_2=3-(-11)=14$; $\Delta x_3=15-11=4$, $\Delta f(x)_3=7-3=4$

Step2: Compute rate of change

$\frac{13}{13}=1$, $\frac{14}{14}=1$, $\frac{4}{4}=1$ (constant)

Problem A2

Step1: Calculate Δx and Δf(x)

$\Delta x_1=4-1=3$, $\Delta f(x)_1=23-(-7)=30$; $\Delta x_2=6-4=2$, $\Delta f(x)_2=63-23=40$; $\Delta x_3=7-6=1$, $\Delta f(x)_3=89-63=26$

Step2: Compute rate of change

$\frac{30}{3}=10$, $\frac{40}{2}=20$, $\frac{26}{1}=26$ (not constant)

Problem A3

Step1: Calculate Δx and Δf(x)

$\Delta x_1=0-(-7)=7$, $\Delta f(x)_1=-3-(-10)=7$; $\Delta x_2=9-0=9$, $\Delta f(x)_2=6-(-3)=9$; $\Delta x_3=12-9=3$, $\Delta f(x)_3=9-6=3$

Step2: Compute rate of change

$\frac{7}{7}=1$, $\frac{9}{9}=1$, $\frac{3}{3}=1$ (constant)

Problem A4

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-6-(-9)=3$, $\Delta f(x)_1=-7-(-7)=0$; $\Delta x_2=-1-(-6)=5$, $\Delta f(x)_2=2-(-7)=9$; $\Delta x_3=14-(-1)=15$, $\Delta f(x)_3=16-2=14$

Step2: Compute rate of change

$\frac{0}{3}=0$, $\frac{9}{5}=1.8$, $\frac{14}{15}\approx0.93$ (not constant)

Problem A5

Step1: Calculate Δx and Δf(x)

$\Delta x_1=2-(-3)=5$, $\Delta f(x)_1=-26-(-21)=-5$; $\Delta x_2=13-2=11$, $\Delta f(x)_2=59-(-26)=85$; $\Delta x_3=17-13=4$, $\Delta f(x)_3=41-59=-18$

Step2: Compute rate of change

$\frac{-5}{5}=-1$, $\frac{85}{11}\approx7.73$, $\frac{-18}{4}=-4.5$ (not constant)

Problem A6

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-10-(-14)=4$, $\Delta f(x)_1=-7-(-11)=4$; $\Delta x_2=-5-(-10)=5$, $\Delta f(x)_2=-2-(-7)=5$; $\Delta x_3=-4-(-5)=1$, $\Delta f(x)_3=-1-(-2)=1$

Step2: Compute rate of change

$\frac{4}{4}=1$, $\frac{5}{5}=1$, $\frac{1}{1}=1$ (constant)

Problem A7

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-14-(-18)=4$, $\Delta f(x)_1=4-2=2$; $\Delta x_2=-12-(-14)=2$, $\Delta f(x)_2=5-4=1$; $\Delta x_3=0-(-12)=12$, $\Delta f(x)_3=11-5=6$

Step2: Compute rate of change

$\frac{2}{4}=0.5$, $\frac{1}{2}=0.5$, $\frac{6}{12}=0.5$ (constant)

Problem A8

Step1: Calculate Δx and Δf(x)

$\Delta x_1=3-(-16)=19$, $\Delta f(x)_1=-11-25=-36$; $\Delta x_2=8-3=5$, $\Delta f(x)_2=44-(-11)=55$; $\Delta x_3=21-8=13$, $\Delta f(x)_3=51-44=7$

Step2: Compute rate of change

$\frac{-36}{19}\approx-1.89$, $\frac{55}{5}=11$, $\frac{7}{13}\approx0.54$ (not constant)
---

Problem B1

Option a

Step1: Calculate Δx and Δf(x)

$\Delta x_1=0-(-2)=2$, $\Delta f(x)_1=-4-0=-4$; $\Delta x_2=6-0=6$, $\Delta f(x)_2=45-(-4)=49$; $\Delta x_3=9-6=3$, $\Delta f(x)_3=96-45=51$

Step2: Compute rate of change

$\frac{-4}{2}=-2$, $\frac{49}{6}\approx8.17$, $\frac{51}{3}=17$ (not constant)

Option b

Step1: Calculate Δx and Δf(x)

$\Delta x_1=7-3=4$, $\Delta f(x)_1=37-17=20$; $\Delta x_2=11-7=4$, $\Delta f(x)_2=57-37=20$; $\Delta x_3=14-11=3$, $\Delta f(x)_3=72-57=15$

Step2: Compute rate of change

$\frac{20}{4}=5$, $\frac{20}{4}=5$, $\frac{15}{3}=5$ (constant)

Option c

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-6-(-7)=1$, $\Delta f(x)_1=16-40=-24$; $\Delta x_2=1-(-6)=7$, $\Delta f(x)_2=0-16=-16$; $\Delta x_3=3-1=2$, $\Delta f(x)_3=7-0=7$

Step2: Compute rate of change

$\frac{-24}{1}=-24$, $\frac{-16}{7}\approx-2.29$, $\frac{7}{2}=3.5$ (not constant)
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Problem B2

Option a

Step1: Calculate Δx and Δf(x)

$\Delta x_1=3-1=2$, $\Delta f(x)_1=-1-(-3)=2$; $\Delta x_2=9-3=6$, $\Delta f(x)_2=5-(-1)=6$; $\Delta x_3=10-9=1$, $\Delta f(x)_3=6-5=1$

Step2: Compute rate of change

$\frac{2}{2}=1$, $\frac{6}{6}=1$, $\frac{1}{1}=1$ (constant)

Option b

Step1: Calculate Δx and Δf(x)

$\Delta x_1=2-(-10)=12$, $\Delta f(x)_1=9-(-51)=60$; $\Delta x_2=15-2=13$, $\Delta f(x)_2=74-9=65$; $\Delta x_3=20-15=5$, $\Delta f(x)_3=99-74=25$

Step2: Compute rate of change

$\frac{60}{12}=5$, $\frac{65}{13}=5$, $\frac{25}{5}=5$ (constant)

Option c

Step1: Calculate Δx and Δf(x)

$\Delta x_1=-3-(-8)=5$, $\Delta f(x)_1=2-65=-63$; $\Delta x_2=4-(-3)=7$, $\Delta f(x)_2=17-2=15$

Step2: Compute rate of change

$\frac{-63}{5}=-12.6$, $\frac{15}{7}\approx2.14$ (not constant)

Answer:

Part A

1. Linear
2. Nonlinear
3. Linear
4. Nonlinear
5. Nonlinear
6. Linear
7. Linear
8. Nonlinear

Part B

1. b) $\boldsymbol{

$$\begin{array}{|c|c|c|c|c|} \hline x & 3 & 7 & 11 & 14 \\ \hline f(x) & 17 & 37 & 57 & 72 \\ \hline \end{array}$$

}$

1. c) $\boldsymbol{

$$\begin{array}{|c|c|c|c|c|} \hline x & -8 & -3 & 4 \\ \hline f(x) & 65 & 2 & 17 \\ \hline \end{array}$$

}$