Question

“The population of the Philippines can be modeled by the function P(x)= 20,000,000e0.0251x where x is the number of years since 1955 (x=0 at year 1955). Assuming that this model is accurate, in what year will the population reach 200 Million?”

4. Which of the following should be determined?

a. P (1955)

b. the value of x

c. p(e)

d. P (2.71828…)



5. The problem can be solved by which of the following?

a. Logarithmic Equations

b. Logarithmic Property

c. Logarithmic Inequality

d. Logarithmic identity



6. In what year will the population reach 200 Million?

a. 2050

b. 2046

c.2038

d. 2020



7. Which of the following notation is read as “h composed with g”?

A. h(g)

B. h ◦ g

C. h• g

D. g ◦ h



8. Given f(x) = 2x and g(x) = x + 5, find (f ◦ g)(x).

A. 2x + 5

B. 2x + 10

C. x + 5

D. x + 10



9. Given f(x) = 4x and g(x) = x – 3, find (f ◦ g)(3).

A. 0

B. 1

C. 2

D. 3











10. Given f(x) = 2x + 1 and g(x) = 3x – 2, find (g ◦ f)(x).

A. 5x – 2

B. 5x + 1

C. 6x – 1

D. 6x + 1



​

Answer 1

“The population of the Philippines can be modeled by the function P(x)= 20,000,000e0.0251x

b. the value of x

a. Logarithmic Equations

c. 2038

B. h ◦ g

B. 2x + 10

D. 3

D. 6x + 1

Reasoning:

• The problem states that P(x) is a function of x, where x is the number of years since 1955. In order to find the year when the population reaches 200 million, we need to determine the value of x that satisfies P(x)=200,000,000.

• To solve for x, we can use logarithmic equations.

• By using the equation P(x)= 20,000,000e0.0251x and solving for x when P(x)=200,000,000, we get x ≈ 20.38 , this corresponds to year 2038

• h ◦ g is the mathematical notation for the composition of functions h and g.

• To find (f ◦ g)(x), we first need to substitute f(x) into g(x) to get      $g(f(x)) = g(2x) = (2x + 5), so (f ◦ g)(x) = g(f(x)) = 2x + 5$

• To find (f ◦ g)(3), we first need to substitute 3 into g(x) to get          g(3) = 3 - 3 = 0, then we substitute that result into f(x) to get $f(g(3)) = f(0) = 4*0 = 0$

• To find (g ◦ f)(x), we first need to substitute f(x) into g(x) to get     $g(f(x)) = g(2x+1) = 3(2x+1)-2 = 6x+1, so (g ◦ f)(x) = g(f(x)) = 6x+1$

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