Find Range and Pre-Image of a Function
(i) Range of \(f\), i.e. \(f(A)\)
(ii) Pre-image of \(6\), \(-3\) and \(5\)
Solution
Given:
$$ f(x)=x^2-2x-3 $$
and
$$ A=\{-2,-1,0,1,2\} $$
Calculate the value of \(f(x)\) for each element of \(A\).
| \(x\) | \(f(x)=x^2-2x-3\) |
|---|---|
| \(-2\) | \(4+4-3=5\) |
| \(-1\) | \(1+2-3=0\) |
| \(0\) | \(0-0-3=-3\) |
| \(1\) | \(1-2-3=-4\) |
| \(2\) | \(4-4-3=-3\) |
Therefore,
$$ f(A)=\{5,0,-3,-4\} $$
Hence, the range of \(f\) is:
$$ \{-4,-3,0,5\} $$
(ii) Pre-images
Pre-image of 6:
No element of \(A\) gives value \(6\).
$$ f^{-1}(6)=\phi $$
Pre-image of \(-3\):
$$ f(0)=-3,\quad f(2)=-3 $$
Therefore,
$$ f^{-1}(-3)=\{0,2\} $$
Pre-image of 5:
$$ f(-2)=5 $$
Therefore,
$$ f^{-1}(5)=\{-2\} $$