y=f (x)=x/1+|x|,x belongs to R , y belongs to R
Answers
Answer:
We have f(x)=1+∣x∣x;x∈R
⇒f(x)={1+xx;x≥01−xx;x≤0}
Answer:
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Class 12
>>Maths
>>Relations and Functions
>>Types of Functions
>>Show that the function f: R→ {x ∈ R : -
Question
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Show that the function f:R→{x∈R:−1<x<1} defined by f(x)=
1+∣x∣
x
,x∈R is one to one and onto function.
Medium
Solution
verified
Verified by Toppr
f(x)=
1−x
x
−1<x<0andf(x)=
1+x
x
,0≤x<1
now(i)f(x)=
1−x
x
−1<x<0
Let f(x
1
)=f(x
2
)
⇒
1−x
1
x
1
=
1−x
2
x
2
⇒x
1
−x
1
x
2
=x
2
−x
1
x
2
⇒x
1
=x
2
⇒f is one-one
Let y=
1−x
x
⇒y−xy=x
⇒y=x+xy
⇒y=x(1+y)
⇒x=
1+y
y
⇒∃x=
1+y
y
for all valves of y
=−1
s.t f(x)=y
⇒f(x) is onto
(ii)f(x)=
1+x
x
0≤x≤1
Let f(x
1
)=f(x
2
)
⇒
1+x
1
x
1
=
1+x
2
x
2
⇒x
1
+x
1
x
2
=x
2
+x
1
x
2
⇒x
1
=x
2
⇒ f is one -one
Let y=
1+x
x
⇒y+xy=x
⇒y=x−xy
⇒y=x(1−y)
⇒x=
1−y
y
⇒ for all values of y
=1∃x=
1−y
y
s.t f(x)=y
⇒ f is onto
hence f(x) is one-one and onto proved