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QuestionEvaluate the linear function g(t)=3t+5g(t) = 3t + 5 for various inputs. Create a table and graph the function. a) Find g(0),g(3),g(1)−g(0),g(2)−g(1),g(1001)−g(1000),g(a+1)−g(a)g(0), g(3), g(1) - g(0), g(2) - g(1), g(1001) - g(1000), g(a+1) - g(a).

Studdy Solution

STEP 1

Assumptions
1. The function given is g(t)=3t+5g(t) = 3t + 5.
2. We need to evaluate the function at specific points.
3. We need to create a table of values for the function.
4. We need to graph the function based on the table of values.
5. We need to calculate the differences between the function's values at specific points.

STEP 2

Create a table of values for the function g(t)=3t+5g(t) = 3t + 5 at selected points. For simplicity, we will choose values of tt from -2 to 3.
tg(t)−23(−2)+5−13(−1)+503(0)+513(1)+523(2)+533(3)+5 \begin{array}{c|c} t & g(t) \\ \hline -2 & 3(-2) + 5 \\ -1 & 3(-1) + 5 \\ 0 & 3(0) + 5 \\ 1 & 3(1) + 5 \\ 2 & 3(2) + 5 \\ 3 & 3(3) + 5 \\ \end{array}

STEP 3

Calculate the values of g(t)g(t) for each tt in the table.
tg(t)−23(−2)+5=−1−13(−1)+5=203(0)+5=513(1)+5=823(2)+5=1133(3)+5=14 \begin{array}{c|c} t & g(t) \\ \hline -2 & 3(-2) + 5 = -1 \\ -1 & 3(-1) + 5 = 2 \\ 0 & 3(0) + 5 = 5 \\ 1 & 3(1) + 5 = 8 \\ 2 & 3(2) + 5 = 11 \\ 3 & 3(3) + 5 = 14 \\ \end{array}

STEP 4

Graph the function using the table of values. Plot the points and draw a straight line through them, as the function is linear.

STEP 5

Determine the value of g(0)g(0) by substituting t=0t = 0 into the function.
g(0)=3(0)+5g(0) = 3(0) + 5

STEP 6

Calculate the value of g(0)g(0).
g(0)=3(0)+5=5g(0) = 3(0) + 5 = 5

STEP 7

Determine the value of g(2)−g(1)g(2) - g(1) by substituting t=2t = 2 and t=1t = 1 into the function and then finding the difference.
g(2)−g(1)=(3(2)+5)−(3(1)+5)g(2) - g(1) = (3(2) + 5) - (3(1) + 5)

STEP 8

Calculate the value of g(2)−g(1)g(2) - g(1).
g(2)−g(1)=(3(2)+5)−(3(1)+5)=(6+5)−(3+5)=11−8=3g(2) - g(1) = (3(2) + 5) - (3(1) + 5) = (6 + 5) - (3 + 5) = 11 - 8 = 3

STEP 9

Determine the value of g(3)g(3) by substituting t=3t = 3 into the function.
g(3)=3(3)+5g(3) = 3(3) + 5

STEP 10

Calculate the value of g(3)g(3).
g(3)=3(3)+5=9+5=14g(3) = 3(3) + 5 = 9 + 5 = 14

STEP 11

Determine the value of g(1)−g(0)g(1) - g(0) by substituting t=1t = 1 and t=0t = 0 into the function and then finding the difference.
g(1)−g(0)=(3(1)+5)−(3(0)+5)g(1) - g(0) = (3(1) + 5) - (3(0) + 5)

STEP 12

Calculate the value of g(1)−g(0)g(1) - g(0).
g(1)−g(0)=(3(1)+5)−(3(0)+5)=(3+5)−(0+5)=8−5=3g(1) - g(0) = (3(1) + 5) - (3(0) + 5) = (3 + 5) - (0 + 5) = 8 - 5 = 3

STEP 13

Determine the value of g(1001)−g(1000)g(1001) - g(1000) by substituting t=1001t = 1001 and t=1000t = 1000 into the function and then finding the difference.
g(1001)−g(1000)=(3(1001)+5)−(3(1000)+5)g(1001) - g(1000) = (3(1001) + 5) - (3(1000) + 5)

STEP 14

Calculate the value of g(1001)−g(1000)g(1001) - g(1000).
g(1001)−g(1000)=(3(1001)+5)−(3(1000)+5)=(3003+5)−(3000+5)=3008−3005=3g(1001) - g(1000) = (3(1001) + 5) - (3(1000) + 5) = (3003 + 5) - (3000 + 5) = 3008 - 3005 = 3

STEP 15

Determine the value of g(a+1)−g(a)g(a+1) - g(a) by substituting t=a+1t = a+1 and t=at = a into the function and then finding the difference.
g(a+1)−g(a)=(3(a+1)+5)−(3(a)+5)g(a+1) - g(a) = (3(a+1) + 5) - (3(a) + 5)

STEP 16

Calculate the value of g(a+1)−g(a)g(a+1) - g(a).
g(a+1)−g(a)=(3(a+1)+5)−(3(a)+5)=(3a+3+5)−(3a+5)=3a+8−3a−5=3g(a+1) - g(a) = (3(a+1) + 5) - (3(a) + 5) = (3a + 3 + 5) - (3a + 5) = 3a + 8 - 3a - 5 = 3
The values are as follows: i) g(0)=5g(0) = 5 ii) g(3)=14g(3) = 14 iii) g(1)−g(0)=3g(1) - g(0) = 3 iv) g(2)−g(1)=3g(2) - g(1) = 3 v) g(1001)−g(1000)=3g(1001) - g(1000) = 3 vi) g(a+1)−g(a)=3g(a+1) - g(a) = 3

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