Question

Find the inverse of each function.

Answer 1

The inverse of a function is the opposite of the function

How to determine the inverse functions

1) y = log2 (3x)

Swap the positions of x and y

$x = \log_2(3y)$

Apply the exponential rule

$2^x = 3y$

Make y the subject

$y = \frac{2^x}3$

Hence, the inverse function is: $y = \frac{2^x}3$

2) y=log3(x-1)

Swap x and y

$x = \log_3(y - 1)$

Apply exponent rule

$3^x = y - 1$

Make y the subject

$y = 3^x + 1$

Hence, the inverse function is: $y = 3^x + 1$

3) y=-2log x

Swap x and y

$x=-2\log y$

Divide both sides by -2

$-0.5x=\log y$

Apply exponent rule

$y = 10^{-0.5x}$

Hence, the inverse function is: $y = 10^{-0.5x}$

4) y=log6(3x)

Swap x and y

$x = \log_6(3y)$

Apply exponent rule

$3y = 6^x$

Make y the subject

$y = \frac{6^x}{3}$

Hence, the inverse function is: $y = \frac{6^x}{3}$

5)y=log3(x+2)

Swap x and y

$x = \log_3(y + 2)$

Apply exponent rule

$y + 2 = 3^x$

Make y the subject

$y = 3^x - 2$

Hence, the inverse function is: $y = 3^x - 2$

6) y=log3(5^3)

Swap x and y

$x = \log_3(5^3)$

Hence, the inverse function is: $x = \log_3(5^3)$

7) y=6^x+5

Swap x and y

$x = 6^y + 5$

Subtract 5 from both sides

$6^y = x - 5$

Apply logarithm

$y = \log_6(x - 5)$

Hence, the inverse function is: $y = \log_6(x - 5)$

9) y=log3 2^x

Swap x and y

$x = \log_3(2^y)$

Apply exponent rule

$2^y = 3^x$

Apply logarithm

$y = \log_2(3^x)$

Hence, the inverse function is: $y = \log_2(3^x)$

10) y=3^x -7

Swap x and y

$x = 3^y - 7$

Add 7 to both sides

$3^y = x + 7$

Apply logarithm

$y = \log_3(x + 7)$

Hence, the inverse function is: $y = \log_3(x + 7)$

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