๐(๐ฅ)=1๐ฅ+5โ1f(x)= x+51โ โ1. Find the inverse of f(x) and its domain.A.๐โ1(๐ฅ)=1๐ฅโ1+5f โ1 (x)= xโ11โ +5, where x โ 1B.๐โ1(๐ฅ)=1๐ฅ+1โ5f โ1 (x)= x+11โ โ5, where x โ โ1C.๐โ1(๐ฅ)=1๐ฅ+1โ5f โ1 (x)= x+11โ โ5, where x โ โ5D.๐โ1(๐ฅ)=1๐ฅโ1+5f โ1 (x)= xโ11โ +5, where x โ โ5SUBMITarrow_backPREVIOUS
Question
๐(๐ฅ)=1๐ฅ+5โ1f(x)= x+51โ โ1. Find the inverse of f(x) and its domain.A.๐โ1(๐ฅ)=1๐ฅโ1+5f โ1 (x)= xโ11โ +5, where x โ 1B.๐โ1(๐ฅ)=1๐ฅ+1โ5f โ1 (x)= x+11โ โ5, where x โ โ1C.๐โ1(๐ฅ)=1๐ฅ+1โ5f โ1 (x)= x+11โ โ5, where x โ โ5D.๐โ1(๐ฅ)=1๐ฅโ1+5f โ1 (x)= xโ11โ +5, where x โ โ5SUBMITarrow_backPREVIOUS
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)(๐ฅ)=1/(๐ฅ+1)โ5, where x โ โ1.
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