Given \( f(x) = 1 + \sqrt{x} \), what are the domain and range of \( f^{-1}(x) \)?
A - HSC - SSCE Mathematics Extension 1 - Question 2 - 2020 - Paper 1
Question 2
Given \( f(x) = 1 + \sqrt{x} \), what are the domain and range of \( f^{-1}(x) \)?
A. \( x \geq 0, \; y \geq 0 \)
B. \( x \geq 0, \; y \geq 1 \)
C. \( x \geq 1, \; y... show full transcript
Worked Solution & Example Answer:Given \( f(x) = 1 + \sqrt{x} \), what are the domain and range of \( f^{-1}(x) \)?
A - HSC - SSCE Mathematics Extension 1 - Question 2 - 2020 - Paper 1
Step 1
Find the Inverse Function
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Answer
To find the inverse function ( f^{-1}(x) ), we start by setting ( y = f(x) = 1 + \sqrt{x} ). Rearranging gives us:
[ y - 1 = \sqrt{x} ]
Next, we square both sides:
[ (y - 1)^2 = x ]
Thus, the inverse function is ( f^{-1}(x) = (x - 1)^2 ).
Step 2
Determine the Domain of the Inverse
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Answer
The domain of the inverse function corresponds to the range of the original function. Since ( f(x) ) starts at 1 when ( x = 0 ) and approaches infinity as ( x ) increases, the range of ( f(x) ) is ( y \geq 1 ). Thus, the domain of ( f^{-1}(x) ) is:
[ x \geq 1 ]
Step 3
Determine the Range of the Inverse
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Answer
The range of the inverse function corresponds to the domain of the original function. Since the function ( f(x) ) takes all non-negative ( x ) values, the range of ( f^{-1}(x) ) is:
[ y \geq 0 ]
