So I figured out how to do the first 6, how do I do the remaining?
7. Evaluate the function below over the domain {-1, 0,
1, 2}. As the values of the domain increase, do the
values of the function increase or decrease?
y = (3/4)ˣ
8. Suppose an investment of $5,000 doubles every 12
years. How much is the investment worth after 36
years? After 48 years?
Write and solve an exponential equation.
9. Does the function represent exponential growth or
exponential decay? Identify the growth or decay
factor.
= 9 ∙ (1/2)ˣ
10. You deposit $520 in an account with 4% interest
compounded monthly. What is the balance in the
account after 5 years?
Is 7 decreasing?
(3/4)^-1 = 4/3
(3/4)^0 = 1
(3/4)^1 = 3/4
(3/4)^2 = 9/16
(3/4)^3 = 27/64
Now is that increasing or decreasing?
To evaluate the function in question 7, you need to substitute the given values of the domain into the function and observe whether the values of the function increase or decrease.
For example, let's evaluate the function for the domain values {-1, 0, 1, 2}:
For x = -1:
y = (3/4)^(-1) = 4/3 ≈ 1.333
For x = 0:
y = (3/4)^0 = 1
For x = 1:
y = (3/4)^1 = 3/4 = 0.75
For x = 2:
y = (3/4)^2 = 9/16 ≈ 0.563
From the evaluated values, you can observe that as the values of the domain increase (going from -1 to 2), the values of the function decrease. Therefore, the function is decreasing as the values of the domain increase.
Moving on to question 8, where you are asked to find the worth of an investment after 36 and 48 years, given that the investment doubles every 12 years. To solve this, you can use the exponential formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal (initial investment)
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = time in years
In this case, the principal (initial investment) is $5,000, and it doubles every 12 years, so the annual interest rate (r) would be 100% or 1. The interest is compounded once per year, so n = 1.
For 36 years:
A = 5000(1 + 1/1)^(1*36) = 5000(2)^3 = 5000 * 8 = $40,000
For 48 years:
A = 5000(1 + 1/1)^(1*48) = 5000(2)^4 = 5000 * 16 = $80,000
Therefore, the investment would be worth $40,000 after 36 years and $80,000 after 48 years.
For question 9, you are given a function = 9 * (1/2)^x. To determine if this function represents exponential growth or decay, you need to examine the base of the exponent (1/2). If the base is greater than 1, it represents exponential growth; if it is between 0 and 1 (exclusive), it represents exponential decay.
Here, the base is 1/2, which is less than 1. Therefore, the function represents exponential decay. The decay factor is the base of the exponent, which is 1/2.
Lastly, question 10 asks you to find the balance in an account after 5 years with $520 deposited and 4% interest compounded monthly. To solve this, you can use the formula for compound interest with monthly compounding:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = time in years
In this case, the principal (initial deposit) is $520, the annual interest rate is 4% or 0.04 (expressed as a decimal), the interest is compounded monthly, so n = 12, and the time is 5 years.
A = 520(1 + 0.04/12)^(12*5) ≈ 520(1.003333)^60 ≈ 520 * 1.185501 ≈ $615.53
Therefore, the balance in the account after 5 years would be approximately $615.53.
Regarding your question about whether 7 is decreasing or not, I assume you are referring to question number 7. As explained earlier, the function in question 7 is decreasing as the values of the domain increase.