Question

Please help!! functions and relations. which pair of functions are inverses of eachother??

Answer 1

GIVEN:

We are given the set of functions as indicated in the attached image.

Required;

To determine which pair of functions are inverses of each other.

Step-by-step solution;

We shall solve each function one after the other to determine which pair are inverses of each other and which pair is not.

Option A:

`\begin{gathered} f(x)=(x)/(8)+3 \\ \\ Re-write\text{ }as\text{ }an\text{ }equation\text{ }with\text{ }respect\text{ }to\text{ }y: \\ \\ y=(x)/(8)+3 \\ \\ Next,\text{ }we\text{ }switch\text{ }places\text{ }for\text{ }x\text{ }and\text{ }y: \\ \\ x=(y)/(8)+3 \\ \\ We\text{ }now\text{ }solve\text{ }for\text{ }y: \\ \\ x-3=(y)/(8) \\ \\ Cross\text{ }multiply: \\ \\ 8(x-3)=y \\ \\ 8x-24=y \\ \\ Therefore,\text{ }we\text{ }now\text{ }write\text{ }in\text{ }function\text{ }notation: \\ \\ f^(-1)(x)=8x-24 \end{gathered}`

Therefore, option A is not a pair of inverse functions.

OptionB;

`\begin{gathered} f(x)=(4)/(x)-3 \\ \\ y=(4)/(x)-3 \\ \\ Switch\text{ }sides\text{ }for\text{ }x\text{ }and\text{ }y; \\ \\ x=(4)/(y)-3 \\ \\ Solve\text{ }for\text{ }y; \\ \\ x+3=(4)/(y) \\ \\ Cross\text{ }multiply: \\ \\ y=(4)/(x+3) \\ \\ Therefore; \\ \\ f^(-1)(x)=(4)/(x+3) \end{gathered}`

Therefore, option B is not a pair of inverse functions.

Option C:

`\begin{gathered} f(x)=10x-5 \\ \\ y=10x-5 \\ \\ Switch\text{ }sides\text{ }for\text{ }x\text{ }and\text{ }y; \\ \\ x=10y-5 \\ \\ Solve\text{ }for\text{ }y; \\ \\ x+5=10y \\ \\ Divide\text{ }both\text{ }sides\text{ }by\text{ }10; \\ \\ (x+5)/(10)=y \\ \\ Therefore: \\ f^(-1)(x)=(x+5)/(10) \end{gathered}`

Therefore, option C is a pair of inverses.

Option D:

`\begin{gathered} f(x)=\sqrt[3]{6x} \\ \\ y=\sqrt[3]{6x} \\ \\ Switch\text{ }sides\text{ }for\text{ }x\text{ }and\text{ }y; \\ \\ x=\sqrt[3]{6y} \\ \\ Solve\text{ }for\text{ }y; \\ \\ Cube\text{ }both\text{ }sides; \\ \\ x^3=(\sqrt[3]{6y})^3 \\ \\ The\text{ }exponent\text{ }cancels\text{ }out\text{ }the\text{ }radical\text{ }on\text{ }the\text{ }right\text{ }side; \\ \\ x^3=6y \\ \\ Divide\text{ }both\text{ }sides\text{ }by\text{ }6; \\ \\ (x^3)/(6)=y \\ \\ Therefore: \\ \\ f^(-1)(x)=(x^3)/(6) \end{gathered}`

Therefore, option D is not a pair of inverses.

ANSWER:

The pair of functions in option C are inverses of each other.