CFE Review Unit 8 and 9

Given the graph of an exponential function that passes through the points (0, 3) and (1, 6), write the exponential function.

Answer explanation

The function is of the form f(x) = a * b^x. Since f(0) = 3, we have a = 3. For f(1) = 6, we get 3 * b = 6, leading to b = 2. Thus, the function is f(x) = 3 * 2^x, which matches the correct choice.

Answer explanation

To find the inverse of the function f(x) = 2x + 3, swap x and y, giving x = 2y + 3. Solving for y, we get y = (x - 3)/2. Thus, the inverse is f^{-1}(x) = (x - 3)/2, which matches the first answer choice.

Answer explanation

To solve the equation \(2^x = 16\), we recognize that \(16\) can be expressed as \(2^4\). Therefore, setting the exponents equal gives us \(x = 4\). This is confirmed since \(2^4 = 16\).

Answer explanation

Using the property of logarithms, \log_b(m) + \log_b(n) = \log_b(m \cdot n), we combine \log_2(8) + \log_2(4) = \log_2(8 \cdot 4) = \log_2(32). Thus, the correct answer is \log_2(32).

• 50%

• 150%

• 75%

• 25%

Answer explanation

The function g(x) = 5 \cdot (1.5)^x has a base of 1.5. The percent rate of change is calculated as (1.5 - 1) \cdot 100\%, which equals 50\%. Thus, the constant percent rate of change is 50\%.

Ethan is saving money in a bank account. He initially deposits $7, and the amount triples every year. Write an exponential function to represent the amount of money in the account after x years.

Answer explanation

Ethan's initial deposit is $7, and it triples each year. The exponential function representing this growth is f(x) = 7 * 3^x, where x is the number of years. Thus, the correct choice is f(x) = 7 * 3^x.

Answer explanation

To find the inverse of the function f(x) = 1/x, we set y = 1/x and solve for x, giving us x = 1/y. Thus, the inverse function is f^{-1}(x) = 1/x, which matches the correct answer.

Answer explanation

To solve \(\log(x) = 2\), we rewrite it in exponential form: \(x = 10^2 = 100\). Thus, the correct answer is \(x = 100\), since \(\log(100) = 2\) confirms the solution.

Answer explanation

Using the property of logarithms, \log_b(a) - \log_b(c) = \log_b(\frac{a}{c}), we simplify \log_3(27) - \log_3(3) = \log_3(\frac{27}{3}) = \log_3(9). Thus, the correct answer is \log_3(9).

• -50%

• 50%

• -100%

• 100%

Answer explanation

The function h(x) = 10 * (0.5)^x shows a decay factor of 0.5. This indicates a constant percent rate of change of -50%, meaning the value decreases by 50% for each unit increase in x.