Find the range and domain of the function :
Answers
Step-by-step explanation:
Hey user;
firstly;
Answer:
Domain is [ 0 , 4 ]
Range is [ 0 , 2 ] .
Step-by-step explanation:
f(x) = √( 4x - x^2 )
For D, f(x) must be a real number. Since d
f(x) = √( 4x - x^2 ), 4x - x^2 must be a positive real number( including 0 ).
Therefore,
= > 4x - x^2 ≥ 0
= > x( 4 - x ) ≥ 0
= > ( x - 0 )( x - 4 ) ≥ 0
= > x ≥ 4 or x ≤ 0
= > 0 ≥ x ≥ 4
Thus, domain of this function is [ 0 , 4 ].
Let √( 4x - x^2 ) = y
= > 4x - x^2 = y^2
= > x^2 - 4x + y^2 = 0
Since x( domain ) is a real number, discriminant of this must be a real number.
= > Discriminant ≥ 0
= > ( - 4 )^2 - 4( 1 )( y^2 ) ≥ 0
= > 16 - 4y^2 ≥ 0
= > 4 - y^2 ≥ 0
= > ( 2 + y )( 2 - y ) ≥ 0
= > - 2 ≥ y or y ≤ 2
y is the square root of 4x - x^2.
y can't be negative, so range is [ 0 , 2 ] .