Answer: C
Step-by-step explanation:
For this problem, let's find the inverse for all f(x) and see which pairs with the g(x). To find the inverse, you replace the x with y and y with x. Then you solve for y.
A. Incorrect
[tex]y=\frac{x-8}{4} +9[/tex] [replace x with y and y with x]
[tex]x=\frac{y-8}{4} +9[/tex] [subtract both sides by 9]
[tex]x-9=\frac{y-8}{4}[/tex] [multiply both sides by 4]
[tex]4(x-9)=y-8[/tex] [add both sides by 8]
[tex]4(x-9)+8=y[/tex]
This does not match g(x), therefore, they are not inverses of each other.
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B. Incorrect
[tex]y=4(x-12)+2[/tex] [replace x with y and y with x]
[tex]x=4(y-12)+2[/tex] [subtract both sides by 2]
[tex]x-2=4(y-12)[/tex] [divide both sides by 4]
[tex]\frac{x-2}{4} =y-12[/tex] [add both sides by 12]
[tex]\frac{x-2}{4} +12=y[/tex]
This does not match g(x), therefore, they are not inverses of each other.
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C. Correct
[tex]y=3(\frac{x}{2})-4[/tex] [replace x with y and y with x]
[tex]x=3(\frac{y}{2} )-4[/tex] [add both sides by 4]
[tex]x+4=3(\frac{y}{2} )[/tex] [divide both sides by 3]
[tex]\frac{x+4}{3} =\frac{y}{2}[/tex] [multiply both sides by 2]
[tex]\frac{2(x+4)}{3} =y[/tex]
This matches g(x), therefore, they are inverses of each other.
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D. Incorrect
[tex]y=3(\frac{2}{x} )-10[/tex] [replace x with y and y with x]
[tex]x=3(\frac{2}{y} )-10[/tex] [add both sides by 10]
[tex]x+10=3(\frac{2}{y} )[/tex] [divide both sides by 3]
[tex]\frac{x+10}{3} =\frac{2}{y}[/tex] [multiply both sides by y]
[tex](y)(\frac{x+10}{3} )=2[/tex] [multiply both sides by [tex]\frac{3}{x+10}[/tex] or divide by [tex]\frac{x+10}{3}[/tex]]
[tex]y=\frac{6}{x+10}[/tex]
This does not match g(x), therefore, they are not inverses of each other.
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After going through each problem, we found that the correct answer is C.
