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The inverse of a function is the opposite of the function
1) y = log2 (3x)
Swap the positions of x and y
[tex]x = \log_2(3y)[/tex]
Apply the exponential rule
[tex]2^x = 3y[/tex]
Make y the subject
[tex]y = \frac{2^x}3[/tex]
Hence, the inverse function is: [tex]y = \frac{2^x}3[/tex]
2) y=log3(x-1)
Swap x and y
[tex]x = \log_3(y - 1)[/tex]
Apply exponent rule
[tex]3^x = y - 1[/tex]
Make y the subject
[tex]y = 3^x + 1[/tex]
Hence, the inverse function is: [tex]y = 3^x + 1[/tex]
3) y=-2log x
Swap x and y
[tex]x=-2\log y[/tex]
Divide both sides by -2
[tex]-0.5x=\log y[/tex]
Apply exponent rule
[tex]y = 10^{-0.5x}[/tex]
Hence, the inverse function is: [tex]y = 10^{-0.5x}[/tex]
4) y=log6(3x)
Swap x and y
[tex]x = \log_6(3y)[/tex]
Apply exponent rule
[tex]3y = 6^x[/tex]
Make y the subject
[tex]y = \frac{6^x}{3}[/tex]
Hence, the inverse function is: [tex]y = \frac{6^x}{3}[/tex]
5)y=log3(x+2)
Swap x and y
[tex]x = \log_3(y + 2)[/tex]
Apply exponent rule
[tex]y + 2 = 3^x[/tex]
Make y the subject
[tex]y = 3^x - 2[/tex]
Hence, the inverse function is: [tex]y = 3^x - 2[/tex]
6) y=log3(5^3)
Swap x and y
[tex]x = \log_3(5^3)[/tex]
Hence, the inverse function is: [tex]x = \log_3(5^3)[/tex]
7) y=6^x+5
Swap x and y
[tex]x = 6^y + 5[/tex]
Subtract 5 from both sides
[tex]6^y = x - 5[/tex]
Apply logarithm
[tex]y = \log_6(x - 5)[/tex]
Hence, the inverse function is: [tex]y = \log_6(x - 5)[/tex]
9) y=log3 2^x
Swap x and y
[tex]x = \log_3(2^y)[/tex]
Apply exponent rule
[tex]2^y = 3^x[/tex]
Apply logarithm
[tex]y = \log_2(3^x)[/tex]
Hence, the inverse function is: [tex]y = \log_2(3^x)[/tex]
10) y=3^x -7
Swap x and y
[tex]x = 3^y - 7[/tex]
Add 7 to both sides
[tex]3^y = x + 7[/tex]
Apply logarithm
[tex]y = \log_3(x + 7)[/tex]
Hence, the inverse function is: [tex]y = \log_3(x + 7)[/tex]
Read more about inverse functions at:
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