Question

multiple choice
identify the choice that best completes the statement or answers the question.

1. use inverse operations to write the inverse of $f(x) = x + \frac{2}{3}$.

a. $f^{-1}(x) = x - \frac{2}{3}$
c. $f^{-1}(x) = x - \frac{1}{3}$
b. $f^{-1}(x) = x + \frac{1}{3}$
d. $f^{-1}(x) = x + \frac{2}{3}$

1. use inverse operations to write the inverse of $f(x) = \frac{x}{4} - 5$.

a. $f^{-1}(x) = -5x - 4$
c. $f^{-1}(x) = \frac{x}{4} + 5$
b. $f^{-1}(x) = 4x + 5$
d. $f^{-1}(x) = 4(x + 5)$

1. write the exponential equation $2^3 = 8$ in logarithmic form.

a. $\log_2 8 = 3$
c. $\log_3 8 = 2$
b. $\log_2 3 = 8$
d. $\log_8 2 = 3$

1. write the logarithmic equation $\log_4 16 = 2$ in exponential form.

a. $2^{-4} = 16$
c. $4^2 = 16$
b. $2^4 = 16$
d. $4^{16} = 2$

Explanation:

Question 1

Step1: Recall inverse function steps

To find the inverse of \( f(x) = x + \frac{2}{3} \), first replace \( f(x) \) with \( y \), so \( y = x + \frac{2}{3} \). Then swap \( x \) and \( y \): \( x = y + \frac{2}{3} \). Now solve for \( y \) by subtracting \( \frac{2}{3} \) from both sides: \( y = x - \frac{2}{3} \), which is \( f^{-1}(x) = x - \frac{2}{3} \).

Step1: Start with \( y = \frac{x}{4} - 5 \)

Replace \( f(x) \) with \( y \): \( y = \frac{x}{4} - 5 \). Swap \( x \) and \( y \): \( x = \frac{y}{4} - 5 \).

Step2: Solve for \( y \)

Add 5 to both sides: \( x + 5 = \frac{y}{4} \). Multiply both sides by 4: \( y = 4(x + 5) \), so \( f^{-1}(x) = 4(x + 5) \).

Step1: Recall exponential to logarithmic conversion

The exponential equation \( a^b = c \) converts to \( \log_a c = b \). For \( 2^3 = 8 \), \( a = 2 \), \( b = 3 \), \( c = 8 \), so \( \log_2 8 = 3 \).

Answer:

a. \( f^{-1}(x) = x - \frac{2}{3} \)

Question 2