Given $f(x) = 1 +
oot{x}$, what are the domain and range of $f^{-1}(x)$?
A - HSC - SSCE Mathematics Extension 1 - Question 2 - 2020 - Paper 1
Question 2
Given $f(x) = 1 +
oot{x}$, 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. $x \geq 1, \; y \geq 0$
D. $x \ge... show full transcript
Worked Solution & Example Answer:Given $f(x) = 1 +
oot{x}$, what are the domain and range of $f^{-1}(x)$?
A - HSC - SSCE Mathematics Extension 1 - Question 2 - 2020 - Paper 1
Step 1
Determine the function and its inverse
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Answer
The function is defined as f(x)=1+x. To find the domain and range of the inverse, we first need to derive the inverse function.
Starting from:
y=1+x
To find the inverse, solve for x:
y−1=x
Squaring both sides gives:
x=(y−1)2
Thus, the inverse function is:
f−1(y)=(y−1)2
Step 2
Find the domain of the inverse function
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Answer
The expression (y−1)2 is defined for all real numbers y. However, since x requires x≥0, we must consider the range of f(x) which is y≥1 after the adjustments with 1. Thus the domain of f−1(y) is:
y≥1
Step 3
Find the range of the inverse function
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Answer
For the inverse function f−1(y)=(y−1)2, since (y−1)2≥0 when y≥1, the range is:
x≥0
