QuestionFind the domain, range, inverse domain, and inverse range of the function .
Studdy Solution
STEP 1
Assumptions1. The function is given by . We are asked to find the domain and range of the function and its inverse.
STEP 2
First, let's find the domain of the function . The domain of a function is the set of all possible input values (x-values) which will output a real number.
For the function , the denominator cannot be zero because division by zero is undefined in mathematics. Therefore, we need to find the x-values for which the denominator is not zero.
So, we set the denominator equal to zero and solve for x.
STEP 3
olving the inequality gives the x-values that are not allowed in the domain.
So, the domain of the function is all real numbers except .
STEP 4
Next, let's find the range of the function . The range of a function is the set of all possible output values (y-values) that we get after substituting all the possible x-values from the domain into the function.
For the function , there is no restriction on the numerator, so it can be any real number. However, as the denominator can be any real number except zero, the function can take any real value except zero.
So, the range of the function is all real numbers except0.
STEP 5
Now, let's find the domain and range of the inverse function. The inverse of a function 'undoes' the operation of the original function.The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.
So, the domain of the inverse function is all real numbers except0, and the range of the inverse function is all real numbers except .
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