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๐‘“(๐‘ฅ)=1๐‘ฅ+5โˆ’1f(x)= x+51โ€‹ โˆ’1. Find the inverse of f(x) and its domain.A.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅโˆ’1+5f โˆ’1 (x)= xโˆ’11โ€‹ +5, where x โ‰  1B.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅ+1โˆ’5f โˆ’1 (x)= x+11โ€‹ โˆ’5, where x โ‰  โ€“1C.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅ+1โˆ’5f โˆ’1 (x)= x+11โ€‹ โˆ’5, where x โ‰  โ€“5D.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅโˆ’1+5f โˆ’1 (x)= xโˆ’11โ€‹ +5, where x โ‰  โ€“5SUBMITarrow_backPREVIOUS

Question

๐‘“(๐‘ฅ)=1๐‘ฅ+5โˆ’1f(x)= x+51โ€‹ โˆ’1. Find the inverse of f(x) and its domain.A.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅโˆ’1+5f โˆ’1 (x)= xโˆ’11โ€‹ +5, where x โ‰  1B.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅ+1โˆ’5f โˆ’1 (x)= x+11โ€‹ โˆ’5, where x โ‰  โ€“1C.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅ+1โˆ’5f โˆ’1 (x)= x+11โ€‹ โˆ’5, where x โ‰  โ€“5D.๐‘“โˆ’1(๐‘ฅ)=1๐‘ฅโˆ’1+5f โˆ’1 (x)= xโˆ’11โ€‹ +5, where x โ‰  โ€“5SUBMITarrow_backPREVIOUS

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Solution

The function given is f(x) = 1/(x+5) - 1. To find the inverse of this function, we first replace f(x) with y to get y = 1/(x+5) - 1.

Step 1: Swap x and y to get x = 1/(y+5) - 1.

Step 2: Solve for y. First, add 1 to both sides to get x + 1 = 1/(y+5).

Step 3: Take the reciprocal of both sides to get (y+5) = 1/(x+1).

Step 4: Finally, subtract 5 from both sides to get y = 1/(x+1) - 5.

So, the inverse function is f^(-1)(x) = 1/(x+1) - 5.

The domain of the inverse function is all real numbers except -1, because x = -1 would make the denominator of the fraction in the inverse function equal to zero, which is undefined. So, the correct answer is B. ๐‘“^(-1)(๐‘ฅ)=1/(๐‘ฅ+1)โˆ’5, where x โ‰  โ€“1.

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